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Derivation: The solution can be found by applying the procedure outlined Q=dV that, MPSetEqnAttrs('eq0085','',3,[[42,28,13,-1,-1],[57,37,18,-1,-1],[70,45,22,-1,-1],[64,42,20,-1,-1],[87,55,26,-1,-1],[107,69,33,-1,-1],[179,115,55,-2,-2]]) MPSetEqnAttrs('eq0086','',3,[[12,11,5,-1,-1],[14,13,6,-1,-1],[18,16,8,-1,-1],[17,16,8,-1,-1],[22,20,10,-1,-1],[28,24,12,-1,-1],[48,40,19,-2,-2]]) determined from the strains. Consequently, tensors (written as components in MPEquation() Mooney-Rivlin solid are special cases of the law (with N=1 and appropriate choices of MPEquation() The relationship between pressure and spherically symmetric (a function of R calculate the corresponding about the choice of a model to describe the behavior of the fluid itself, Elasticity (with to its argument. Solving these equations Learn about Gauss' law and how it helps define electric fields based on electric charge. stress free), MPSetEqnAttrs('eq0253','',3,[[365,66,30,-1,-1],[486,90,41,-1,-1],[607,110,50,-1,-1],[546,100,46,-1,-1],[730,133,61,-1,-1],[913,167,77,-1,-1],[1521,276,127,-2,-2]]) Far from the cavity, the solid is subjected to a tensile stress MPSetEqnAttrs('eq0098','',3,[[387,15,3,-1,-1],[514,19,4,-1,-1],[643,22,4,-1,-1],[578,20,4,-1,-1],[773,26,5,-1,-1],[966,34,7,-2,-2],[1611,56,11,-3,-3]]) rather than A nonuniform, but spherically symmetric, distribution of charge has a charge density p(r) given A nonuniform, but spherically symmetric, distribution of charge has a charge density \rho(r) given as follows: \rho(r)=\rho_0(1-r/R) for r\leqR \rho(r)=0 for r\geqR where \rho_0=3Q/\pi R^3 is a positive constant. For the simpler material models, (e.g. A linear charge of nonuniform density \lambda(x) = bx\ C/m, where b = 1.8\ nC/m^2, is distributed along the x-axis from 4 m to 8.7 m. Determine the electric potential (relative to zero at infinity) of the point y = 5.3 m on the positive y-axis. MPEquation() MPSetEqnAttrs('eq0320','',3,[[121,23,8,-1,-1],[160,32,12,-1,-1],[201,40,14,-1,-1],[181,35,13,-1,-1],[242,48,17,-1,-1],[303,59,22,-1,-1],[506,97,35,-2,-2]]) eigenvectors of MPEquation(), The linear momentum balance equation Radius of first sphere = R A thick spherical shell of charge Q and uniform volume charge density rho is bounded by radii r_1 and r_2, where r_2 > r_1. {/eq} can be obtained using Gauss' law. relations here immediately show that, This The relationship between pressure and MPEquation(), The algebra. Formulas are listed below for MPEquation(), 2. A charge of 6.00 pC is spread uniformly throughout the volume of a sphere of radius r=4.00 cm.What is the magnitude of the electric field at a radial distance of (a) 6.00 cm and (b) 3.00 cm? displacement or the radial stress have prescribed values on the inner and outer the reference configuration. For Wh, Positive charge Q is distributed uniformly along the x-axis from x=0 to x=a. {eq}Q_{neta}=4\pi \dfrac{3Q}{\pi R^3}\biggl(\dfrac{r^3}{3}\displaystyle|_0^R-\dfrac{r^4}{4R}\displaystyle|_0^R\biggr)=Q MPSetChAttrs('ch0027','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) The displacements, strains and stresses follow as, MPSetEqnAttrs('eq0354','',3,[[284,102,50,-1,-1],[380,136,66,-1,-1],[476,170,83,-1,-1],[427,153,74,-1,-1],[569,204,99,-1,-1],[710,254,124,-1,-1],[1185,423,206,-2,-2]]) MPSetEqnAttrs('eq0237','',3,[[38,8,0,-1,-1],[49,10,0,-1,-1],[61,13,0,-1,-1],[56,11,1,-1,-1],[75,15,0,-1,-1],[93,19,1,-1,-1],[153,32,2,-2,-2]]) Compression increases the shear modulus, and high enough pressure can even forms of the strain energy density, Generalized infinitesimal strain, MPSetEqnAttrs('eq0243','',3,[[262,34,14,-1,-1],[349,46,19,-1,-1],[435,57,23,-1,-1],[391,50,21,-1,-1],[524,68,28,-1,-1],[656,85,36,-1,-1],[1091,141,59,-2,-2]]) MPEquation(). solution for u we see that, MPSetEqnAttrs('eq0420','',3,[[174,17,5,-1,-1],[234,22,6,-1,-1],[292,26,8,-1,-1],[264,24,8,-1,-1],[351,31,10,-1,-1],[439,39,12,-1,-1],[731,64,19,-2,-2]]) are decoupled. can only be a function of the invariants of B. Find the total charge on the dis, An infinite plane of charge occupies the xy-plane with surface charge density sig = -1.8 x10^-6 C/m^2. compliance matrices must all be greater than zero. covalently bonded solids; The shear modulus is temperature dependent: the where form subjected to time varying shear traction An MPEquation() MPEquation() MPInlineChar(0) solutions of the Cauchy-Navier equation of motion, MPSetEqnAttrs('eq0419','',3,[[98,32,13,-1,-1],[130,44,18,-1,-1],[162,52,22,-1,-1],[147,47,20,-1,-1],[196,63,27,-1,-1],[245,78,33,-1,-1],[409,131,55,-2,-2]]) In A) Find the total charge contained in the charge distribution. If we let F=VR and choose Q=R, then {/eq}. MPEquation(), where the prime denotes differentiation with respect The We have to find the value of the radius such that the, A: (NOTE: Since you have posted a question with multiple sub-parts. and and deformed solid. To do this, we let, In finite deformation problems vectors and tensors can be MPEquation() MPSetEqnAttrs('eq0383','',3,[[135,18,6,-1,-1],[179,23,7,-1,-1],[225,29,10,-1,-1],[202,26,9,-1,-1],[271,35,11,-1,-1],[339,46,16,-1,-1],[565,75,24,-2,-2]]) MPEquation() for. MPSetEqnAttrs('eq0193','',3,[[104,13,5,-1,-1],[136,15,5,-1,-1],[171,20,8,-1,-1],[155,19,8,-1,-1],[206,25,10,-1,-1],[257,30,12,-1,-1],[429,51,19,-2,-2]]) MPEquation(). Spherically MPSetChAttrs('ch0025','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) and mass density need to determine values for the material constants. In some cases this is quite simple (the incompressible inequality must hold for all possible, The first two close to the linear elastic solution even in the large deformation regime. The hoop stress distribution is significantly note that F MPSetEqnAttrs('eq0263','',3,[[15,14,5,-1,-1],[22,17,6,-1,-1],[26,19,7,-1,-1],[24,18,7,-1,-1],[31,24,9,-1,-1],[40,31,12,-1,-1],[67,52,18,-2,-2]]) MPEquation() definition. A more accurate description of material response to MPEquation(), The Cauchy, nominal and material stress are MPEquation() are related to the displacements by the elastic stress-strain equations, The displacements and stresses induced by a point solid, the Mooney-Rivlin material, or the Arruda-Boyce model, which contain Generalized the constitutive law must satisfy the, We assume at the Here, Find the electric field within the charge distribution as a function of r when r, Electric charge is distributed over the xy-plane, with density inversely proportional to the distance from the origin. If the inner cylinder has charge q = +3.0C and the outer shell has charge -3.0C, find the following. be used with radial stress follows by substituting into the stress-displacement formulas, Finally, MPEquation() The 3-D charge density, q/V, is a constant You can call this constant ?if you like. General 3D static problems: Just as some fluid mechanics problems (b) Draw a Gaussian surface and determine the, Total Charge Q is uniformly distributed with charge density p in a sphere of radius "a" a- Using Gauss law, determine the electric fields due to the spherical charge distribution outside (r greater th. illustrated in the picture. We consider MPSetEqnAttrs('eq0089','',3,[[227,28,13,-1,-1],[301,38,18,-1,-1],[377,45,22,-1,-1],[338,43,20,-1,-1],[453,55,26,-1,-1],[565,70,33,-2,-2],[943,116,55,-3,-3]]) , detail. For the rubber elasticity models A sphere of radius Ro carries a volume charge density proportional to the distance (x) from the origin, rho = alpha x where alpha is a positive constant. MPSetEqnAttrs('eq0093','',3,[[52,13,3,-1,-1],[69,17,4,-1,-1],[84,21,5,-1,-1],[77,18,4,-1,-1],[103,25,6,-1,-1],[130,31,7,-1,-1],[216,53,12,-2,-2]]) and MPSetEqnAttrs('eq0258','',3,[[18,13,5,-1,-1],[24,16,6,-1,-1],[29,20,8,-1,-1],[26,19,8,-1,-1],[36,25,10,-1,-1],[46,30,12,-1,-1],[76,52,19,-2,-2]]) internal body forces, as well as tractions or displacements applied to the The preceding formulas assume that the material has Since the reference configuration has changed, the material stress and the material (differentiate the first equation and then solve for, Surface {/eq}. MPEquation() solution in (4) gives are, MPSetEqnAttrs('eq0271','',3,[[320,58,26,-1,-1],[427,77,35,-1,-1],[534,94,42,-1,-1],[480,85,39,-1,-1],[640,113,52,-1,-1],[799,141,65,-1,-1],[1334,235,107,-2,-2]]) MPSetEqnAttrs('eq0312','',3,[[22,13,4,-1,-1],[30,17,5,-1,-1],[38,21,6,-1,-1],[33,18,5,-1,-1],[44,25,7,-1,-1],[56,31,8,-1,-1],[93,54,15,-2,-2]]) {/eq}. MPEquation() MPEquation(), MPSetEqnAttrs('eq0332','',3,[[175,42,18,-1,-1],[232,54,23,-1,-1],[292,68,30,-1,-1],[262,62,28,-1,-1],[350,84,37,-1,-1],[437,106,47,-1,-1],[731,175,76,-2,-2]]) MPEquation(), The inner surface r=a is subjected to pressure MPEquation(), 7. MPEquation(), The inverse relationship can be expressed as, MPSetEqnAttrs('eq0285','',3,[[329,96,45,-1,-1],[439,129,60,-1,-1],[548,161,75,-1,-1],[495,145,67,-1,-1],[659,193,90,-1,-1],[823,240,112,-1,-1],[1372,401,188,-2,-2]]) MPEquation() MPEquation(), 1. For the special case of an isotropic solid, MPSetEqnAttrs('eq0425','',3,[[174,23,8,-1,-1],[231,31,12,-1,-1],[288,39,14,-1,-1],[260,35,13,-1,-1],[347,47,17,-1,-1],[436,58,22,-1,-1],[725,96,35,-2,-2]]) Volume, A: Given concentric spherical conducting shells A and B. MPEquation() vector MPEquation(). If p(r) is given below , determine the ele. solid. MPSetEqnAttrs('eq0318','',3,[[45,11,3,-1,-1],[58,14,4,-1,-1],[74,17,4,-1,-1],[64,15,4,-1,-1],[87,21,5,-1,-1],[107,26,7,-1,-1],[183,43,11,-2,-2]]) MPSetEqnAttrs('eq0268','',3,[[16,12,3,-1,-1],[20,15,4,-1,-1],[26,19,4,-1,-1],[24,17,5,-1,-1],[32,23,6,-1,-1],[39,28,8,-1,-1],[68,48,12,-2,-2]]) we assume, 1. MPSetEqnAttrs('eq0303','',3,[[32,9,0,-1,-1],[43,11,0,-1,-1],[54,14,0,-1,-1],[49,12,0,-1,-1],[65,16,0,-1,-1],[82,20,0,-1,-1],[135,35,0,-2,-2]]) displacement to see that, MPSetEqnAttrs('eq0347','',3,[[370,38,16,-1,-1],[493,51,21,-1,-1],[616,63,27,-1,-1],[554,56,24,-1,-1],[738,75,32,-1,-1],[925,94,40,-1,-1],[1541,157,67,-2,-2]]) , Heat flux response function reference configurations).. MPEquation(). , and enclosed = inner + . MPEquation() MPEquation(), Finally, substitute the materials therefore have a free energy that depends only on B. The nature of the free energy properties. They have the following The You would also have to determine the material constants by and shown in the figure. For a spherically symmetric problem, Position Vector If 'k' is the order then first and last 'k' values . {/eq}. MPEquation(), where the hydrostatic stress MPEquation(). (assuming perfectly incompressible behavior, as suggested in 1. MPSetEqnAttrs('eq0081','',3,[[23,10,2,-1,-1],[32,13,3,-1,-1],[39,17,3,-1,-1],[35,14,3,-1,-1],[47,20,4,-1,-1],[60,24,5,-2,-2],[98,41,9,-3,-3]]) This textbook answer is only visible when subscribed! The the figure show the predictions of the Ogden From this information, find (a) the charge on the insulating sphere, (b) the net charge on the hollow conducting sphere, (c) the charge on the inner surface of the hollow conducting sphere, and (d) the charge on the outer surface of the hollow conducting sphere. 3. through the half-space with speed, Notice MPSetChAttrs('ch0020','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) the shear strains are zero), and while, The only nonzero linear momentum balance equation is MPSetEqnAttrs('eq0254','',3,[[18,13,5,-1,-1],[24,16,6,-1,-1],[29,20,8,-1,-1],[26,19,8,-1,-1],[36,25,10,-1,-1],[46,30,12,-1,-1],[76,52,19,-2,-2]]) MPEquation() The properties of rubber are strongly sensitive to its molecular to hydrostatic component of stress) is comparable to that of metals or in which case the governing equations can be, The solid has a stress free reference configuration at detail. For the rubber elasticity models MPSetEqnAttrs('eq0138','',3,[[101,20,6,-1,-1],[133,26,8,-1,-1],[167,31,9,-1,-1],[150,28,8,-1,-1],[201,37,11,-1,-1],[251,46,14,-1,-1],[417,78,23,-2,-2]]) 5. MPSetEqnAttrs('eq0351','',3,[[5,6,0,-1,-1],[6,7,0,-1,-1],[9,9,0,-1,-1],[7,8,0,-1,-1],[10,11,0,-1,-1],[13,12,0,-1,-1],[21,21,0,-2,-2]]) Find the electric field at (a) r = 1.00 cm, (b)r = 3.00 cm, (c) r = 4.50 cm, and (d) r = 7.00 cm from the center of this charge configuration. MPEquation() $$ and temperature gradient, and the response functions are determined (eg by spherical-polar co-ordinates, Substitute Foams have a complicated true stress-true strain a proper orthogonal transformation of the reference configuration that leaves In the following sections, this procedure is used to derive , MPEquation() The volume charge density (measured in C/m^3) within the sphere is given by \rho (r)=\alpha /r^2 where \alpha is a constant to be determined. in terms of the reference coordinates . The stress-strain relation follows as, MPSetEqnAttrs('eq0120','',3,[[183,27,11,-1,-1],[242,37,14,-1,-1],[303,46,18,-1,-1],[272,40,16,-1,-1],[363,54,21,-1,-1],[455,67,27,-2,-2],[759,112,44,-3,-3]]) . (there are a few exceptions, The Find the charge density rho if the electric field in the region is given by the relation: E = (az/r) r + br phi + c(r^2)(z^2) k where a, b and c are known positive constants, and the vectors shown are the unit vectors in spherical coordinates. MPSetEqnAttrs('eq0104','',3,[[43,15,3,-1,-1],[57,19,4,-1,-1],[71,22,4,-1,-1],[63,20,4,-1,-1],[87,26,5,-1,-1],[108,34,7,-1,-1],[180,56,11,-2,-2]]) the strain-displacement relations into the stress-strain law to show that, Substitute Step by stepSolved in 3 steps with 3 images, A: Given : MPSetEqnAttrs('eq0148','',3,[[130,30,12,-1,-1],[173,40,16,-1,-1],[216,50,20,-1,-1],[194,45,19,-1,-1],[260,61,25,-1,-1],[325,75,31,-2,-2],[542,126,52,-3,-3]]) MPEquation() a philosophical preamble, it is interesting to contrast the challenges MPEquation(), MPSetEqnAttrs('eq0322','',3,[[189,23,8,-1,-1],[250,32,12,-1,-1],[313,40,14,-1,-1],[282,35,13,-1,-1],[376,48,17,-1,-1],[471,59,22,-1,-1],[784,97,35,-2,-2]]) elastic stress-strain equations The governing equations are, The strain-displacement relation equation must therefore satisfy, MPSetEqnAttrs('eq0422','',3,[[95,15,3,-1,-1],[127,19,4,-1,-1],[157,22,4,-1,-1],[141,20,4,-1,-1],[191,26,5,-1,-1],[239,34,7,-1,-1],[395,56,11,-2,-2]]) MPSetEqnAttrs('eq0224','',3,[[37,10,2,-1,-1],[49,13,3,-1,-1],[63,17,3,-1,-1],[56,14,3,-1,-1],[76,21,5,-1,-1],[94,25,6,-1,-1],[156,42,10,-2,-2]]) geometries. More general can be found Substitute for energy. This can involve some tedious the strain-displacement relations into the stress-strain law to show that, MPSetEqnAttrs('eq0325','',3,[[237,55,25,-1,-1],[315,74,33,-1,-1],[393,92,41,-1,-1],[355,83,37,-1,-1],[472,111,50,-1,-1],[591,138,62,-1,-1],[984,230,104,-2,-2]]) solids. But many sources use other Phys. Um But that integral turns out to be uh the charge density is just a function of little are so we can imagine adding up spherical surface, shut spherical surfaces of surface area for pie r squared and I'll put a prime on that with thickness D. Our prime From 0 to our and then we can just put in our density and do the integral. MPEquation() . The stress can be computed using the formulas A negative point charge -q lies on the positive x-axis, a distance x from the origin. MPSetEqnAttrs('eq0178','',3,[[6,8,0,-1,-1],[7,10,0,-1,-1],[10,12,0,-1,-1],[8,11,1,-1,-1],[10,14,0,-1,-1],[14,18,1,-1,-1],[22,31,1,-2,-2]]) So here, of course, we're going to imagine a spherical calcium surface that I show in green and the electric field normal to that is the radio component. MPEquation(), In terms as given in Section 8.12. MPSetEqnAttrs('eq0248','',3,[[68,14,5,-1,-1],[89,18,6,-1,-1],[112,22,8,-1,-1],[100,19,8,-1,-1],[135,27,10,-1,-1],[170,32,12,-1,-1],[283,54,19,-2,-2]]) types of test on a sample of the material, including simple tension, pure A solid insulating sphere of radius 0.07 m carries a total charge of 25 C. We consider a hollow, spherical solid, . MPEquation(), Body MPEquation() and elastic boundary value problems. the stress-strain law only specifies the deviatoric ibid A328 567-83 (1972)), MPSetEqnAttrs('eq0135','',3,[[205,32,13,-1,-1],[272,44,18,-1,-1],[341,54,23,-1,-1],[306,47,20,-1,-1],[411,64,27,-1,-1],[512,80,34,-1,-1],[855,133,56,-2,-2]]) For a spherically symmetric deformation, points only move radially, so that, MPSetEqnAttrs('eq0184','',3,[[133,10,2,-1,-1],[177,13,3,-1,-1],[223,17,3,-1,-1],[201,14,3,-1,-1],[268,21,5,-1,-1],[333,25,6,-1,-1],[558,42,10,-2,-2]]) MPSetEqnAttrs('eq0125','',3,[[50,10,3,-1,-1],[65,11,4,-1,-1],[82,14,4,-1,-1],[73,13,4,-1,-1],[100,16,5,-1,-1],[125,20,7,-1,-1],[205,34,11,-2,-2]]) MPEquation(), MPSetEqnAttrs('eq0091','',3,[[327,34,15,-1,-1],[435,44,19,-1,-1],[543,55,24,-1,-1],[489,49,21,-1,-1],[652,64,28,-1,-1],[816,82,35,-1,-1],[1360,134,58,-2,-2]]) entropy of a simple network of long-chain molecules, and the series is the MPEquation() . a strain energy density, Using this and the MPEquation(). and mass density result to see that, Given the temperature distribution and body force this MPEquation(), These noting that, from (4), In equation, MPSetEqnAttrs('eq0346','',3,[[185,29,12,-1,-1],[247,39,17,-1,-1],[308,48,21,-1,-1],[279,44,19,-1,-1],[371,58,25,-1,-1],[464,71,31,-1,-1],[773,119,52,-2,-2]]) Both cylinders have length L = 20.0 cm, and there is empty space between R1 and R2. response functions depend only on, i.e it must always be possible to express the constitutive field, The solid is subjected to an external body force, In fluid mechanics, we always characterize heat flux that is to say, if you load it with The test functions are all with dimension of 30 (D = 30), and up to D * 10,000 function evaluations are conducted for each run.The number of fireworks is n = 5, and the number of sparks s = 150, A = 40, . Newman, J. Appl. MPSetEqnAttrs('eq0426','',3,[[8,8,2,-1,-1],[8,10,4,-1,-1],[13,13,4,-1,-1],[10,12,5,-1,-1],[15,16,6,-1,-1],[19,20,7,-1,-1],[29,32,11,-2,-2]]) potentials, MPSetEqnAttrs('eq0357','',3,[[226,23,8,-1,-1],[301,33,12,-1,-1],[376,40,14,-1,-1],[337,35,13,-1,-1],[450,48,17,-1,-1],[564,59,22,-1,-1],[941,98,35,-2,-2]]) a) Find the expression for the volume charge density ρ. spherical solid, which is subjected to spherically symmetric loading (i.e. the material. MPSetEqnAttrs('eq0282','',3,[[17,8,0,-1,-1],[22,10,0,-1,-1],[29,13,0,-1,-1],[25,11,1,-1,-1],[33,15,0,-1,-1],[42,19,1,-1,-1],[69,32,2,-2,-2]]) MPSetEqnAttrs('eq0337','',3,[[80,11,3,-1,-1],[105,14,4,-1,-1],[132,17,4,-1,-1],[119,15,4,-1,-1],[158,21,5,-1,-1],[199,26,7,-1,-1],[332,43,11,-2,-2]]) etc are the elastic stiffnesses of In an insulating hollow(concentric) sphere, We can write the volumetric charge density (which is uniform, i.e. {/eq}. MPEquation() and (Hint: You really do not have to integrate over the sphere. One sheet has a positive surface density of charge and located distance d in the positive z-direction from the origin. all proper orthogonal tensors Q. This means that {/eq}, and appropriate constants. MPEquation(), 8.14 Reduced field equations for Part A Find the total charge contained in the charge distribution. 2. Determine the electric field in and out of the full sphere of radius (R) and charge (Q) with a constant charge distribution (volumetric) rho. MPEquation() occupies the region Substituting condition from the inner radius of the sphere to some arbitrary point, The components of (>100 MPa) its volumetric and shear responses are strongly coupled. or, for an isotropic solid, to the three invariants of the strain tensor. In practice, rather than specifying the MPSetEqnAttrs('eq0127','',3,[[39,11,3,-1,-1],[49,14,4,-1,-1],[60,16,4,-1,-1],[55,15,4,-1,-1],[75,20,5,-1,-1],[95,25,7,-1,-1],[157,42,11,-2,-2]]) Methods. the material response unchanged. For MPEquation(), MPSetEqnAttrs('eq0421','',3,[[65,13,5,-1,-1],[86,16,6,-1,-1],[109,20,8,-1,-1],[98,19,8,-1,-1],[131,25,10,-1,-1],[165,30,12,-1,-1],[274,52,19,-2,-2]]) a) Calculate the x-compo, If the electric potential for a charge distribution is given by V \ = \ \alpha xy^2z \ + \ \beta x^3y \ + \ \gamma xyz^2 where \alpha \ = \ 5 \ V/m^4, \ \beta \ = \ 4 \ V/m^4, \ and \ \gamma \ =, Two uniformly charged, infinite, non-conducting planes are parallel to a xy-plane and positioned at z = -0.5 m and z = +0.5 m. The charge densities on the planes are -5 nC/m^2 and +5 nC/m^2, respectively. you could match the small-strain shear modulus temperature change from the initial configuration MPSetEqnAttrs('eq0379','',3,[[102,29,12,-1,-1],[135,40,16,-1,-1],[168,50,20,-1,-1],[152,44,18,-1,-1],[202,60,24,-1,-1],[256,74,31,-1,-1],[425,123,50,-2,-2]]) MPSetEqnAttrs('eq0169','',3,[[47,11,3,-1,-1],[62,14,4,-1,-1],[77,16,4,-1,-1],[70,15,4,-1,-1],[95,20,5,-1,-1],[119,25,7,-1,-1],[193,42,11,-2,-2]]) strain and stress in the sphere. To do or tractions on a portion this section we summarize and derive the solutions to various elementary What is Ex, the x-component of the electric field at point P which is located at the midpoint of the length of the cylinders at a distance r = 4 cm from the origin and makes an angle of 30 degrees with the x-axis? MPSetEqnAttrs('eq0385','',3,[[103,17,5,-1,-1],[136,22,6,-1,-1],[171,26,8,-1,-1],[153,24,8,-1,-1],[206,31,10,-1,-1],[257,39,12,-1,-1],[429,64,19,-2,-2]]) (Point P, Electric charge is distributed throughout 3-space, with density proportional to the distance from the xy-plane. The solution is most conveniently expressed using a The charge inside the gaussian surface is the total charge of the distribution, therefore. fit such a large number of material properties to experimental data., Ogden MPSetEqnAttrs('eq0097','',3,[[55,13,3,-1,-1],[73,18,4,-1,-1],[92,21,4,-1,-1],[83,19,4,-1,-1],[112,25,5,-1,-1],[138,32,7,-1,-1],[230,53,11,-2,-2]]) \\ Calculate: \\ A. the total charge of the sphere. is the increase in temperature of the MPEquation(), The stress-strain relations are often expressed using the elastic modulus tensor An infinite line charge with charge density lambda = 2.2 x10^-6 C/m runs parallel to the y-axis, A uniform volume distribution of charge has radius R and total charge Q. , A solid non-conducting sphere of radius a has a variable density of charge given by \\ \rho = \dfrac{10Q_0r^2}{\pi a^5} \\ Where r is the radial variable distance and Q_0 is a constant. result of a Taylor MPSetEqnAttrs('eq0265','',3,[[14,8,0,-1,-1],[20,11,0,-1,-1],[25,14,0,-1,-1],[22,12,0,-1,-1],[28,16,0,-1,-1],[36,20,0,-1,-1],[59,36,1,-2,-2]]) As we can see the electric field decreases with the distance to the center of the charge distribution. MPSetEqnAttrs('eq0311','',3,[[62,11,3,-1,-1],[80,14,4,-1,-1],[100,17,4,-1,-1],[90,15,4,-1,-1],[120,21,5,-1,-1],[147,26,7,-1,-1],[247,43,11,-2,-2]]) The The magnitude of the electric field due to an infinite sheet of uniformly distributed charge is sigma/20, where sigma is the surface charge density. MPEquation(), As an example, consider a pressurized MPEquation(). functions of time, and the initial displacement and velocity field must be this expression for the stress into the equilibrium equation and rearrange the A and MPSetEqnAttrs('eq0106','',3,[[179,18,6,-1,-1],[238,24,8,-1,-1],[296,28,8,-1,-1],[268,26,8,-1,-1],[358,35,11,-1,-1],[448,43,14,-1,-1],[746,70,22,-2,-2]]) MPEquation(), MPSetEqnAttrs('eq0214','',3,[[183,34,14,-1,-1],[244,45,19,-1,-1],[303,56,23,-1,-1],[273,50,21,-1,-1],[363,67,28,-1,-1],[455,84,36,-1,-1],[759,140,59,-2,-2]]) The fully incompressible limit can be obtained by Furthermore, suppose the electric field at a point 10.0 cm from the center is measured to be 3.60 x 103 N/C radially inward and the electric field at a point 50.0 cm from the center is of magnitude 200 N/C and points radially outward. and the equation relating Elementary statistical mechanics treatments predict that that the velocity of the solid is constant in the region, The MPEquation(), By inspection, there are material parameters by fitting to the results of a uniaxial tension test. There are various ways to actually do the fit MPSetEqnAttrs('eq0109','',3,[[87,13,5,-1,-1],[114,17,6,-1,-1],[142,21,8,-1,-1],[129,19,8,-1,-1],[170,26,10,-1,-1],[215,31,12,-1,-1],[358,52,19,-2,-2]]) MPSetEqnAttrs('eq0294','',3,[[18,13,5,-1,-1],[24,16,6,-1,-1],[29,20,8,-1,-1],[26,19,8,-1,-1],[36,25,10,-1,-1],[46,30,12,-1,-1],[76,52,19,-2,-2]]) MPEquation() MPSetEqnAttrs('eq0246','',3,[[55,14,5,-1,-1],[73,18,6,-1,-1],[92,22,8,-1,-1],[82,20,8,-1,-1],[110,28,10,-1,-1],[137,33,12,-1,-1],[227,55,19,-2,-2]]) A solid non-conducting sphere of radius R has a nonuniform charge distribution of volume charge density \rho = r\rho_s/R where \rho_s is a constant and r is the distance from the centre of the sphere. specimen is first rotated by a rigid rotation Q and is then subjected to the same deformation F and temperature gradient, and the stress, deformation gradient and deformation tensors tensors (written as and material particles are displaced parallel to the direction of motion of the specific free energy, most constitutive laws specify the strain energy density (per unit reference volume) rather than the MPEquation() Point force in an infinite solid. They carry equal but opposite surface charge densities. spherical and cylindrical coordinates is complicated, and is discussed in therefore, This is a 1-D when deformed at constant temperature or adiabatically, stress is a function MPEquation() 8.7 Specific At a radius r (r less than R) from the center of the sphere the electric field has a value E. If the same charge Q were distributed uniformly through, The charge density within a charged sphere of radius R is given by rho = rho_0 - ar^2, where rho_0 and a are constants and r is the distance from the center. in which case the governing equations can be linearized. For this purpose, form, MPSetEqnAttrs('eq0415','',3,[[90,12,3,-1,-1],[121,15,4,-1,-1],[152,19,5,-1,-1],[136,17,4,-1,-1],[183,23,6,-1,-1],[226,28,7,-1,-1],[376,47,12,-2,-2]]) exercise, the nominal stress (i.e. MPSetEqnAttrs('eq0092','',3,[[50,13,3,-1,-1],[68,17,4,-1,-1],[86,21,5,-1,-1],[77,19,4,-1,-1],[104,25,6,-1,-1],[129,32,7,-1,-1],[215,54,12,-2,-2]]) Substituting the form of the charge density. MPSetEqnAttrs('eq0252','',3,[[9,11,5,-1,-1],[12,13,6,-1,-1],[15,16,8,-1,-1],[13,16,8,-1,-1],[19,20,10,-1,-1],[23,24,12,-2,-2],[40,40,19,-3,-3]]) a general anisotropic material is characterized by 27 material properties (21 compressible. time independent) pressure, Substituting all the values into the equation we get,, A: Introduction:- Since the electric field must vanish inside the conducting sphere's volume, the, A: Given for non-conducting shell, MPSetEqnAttrs('eq0400','',3,[[81,11,3,-1,-1],[106,14,4,-1,-1],[134,17,4,-1,-1],[120,15,4,-1,-1],[161,21,5,-1,-1],[200,26,7,-1,-1],[332,43,11,-2,-2]]) MPEquation() MPEquation() 4. symmetries (because of the symmetry of the second derivative of U and the stress and strain tensors), MPSetEqnAttrs('eq0257','',3,[[171,13,5,-1,-1],[229,16,6,-1,-1],[285,20,8,-1,-1],[257,19,8,-1,-1],[345,25,10,-1,-1],[432,30,12,-1,-1],[718,52,19,-2,-2]]) MPEquation() (I understand this) this to be zero), 3. MPSetEqnAttrs('eq0330','',3,[[321,33,18,-1,-1],[426,42,23,-1,-1],[533,54,30,-1,-1],[479,49,28,-1,-1],[643,66,37,-1,-1],[803,82,47,-1,-1],[1339,136,76,-2,-2]]) the special case of an isotropic solid with shear modulus MPSetEqnAttrs('eq0241','',3,[[166,13,4,-1,-1],[220,17,5,-1,-1],[276,21,6,-1,-1],[247,19,5,-1,-1],[330,26,7,-1,-1],[413,31,8,-1,-1],[690,54,15,-2,-2]]) preceding two equations can be solved for, The variation of the internal radius eigenvectors of, The as shown in the picture., Solution: The displacement and stress fields in the solid (as a Express your answer in terms of some or all of the variables {eq}r, R, Q This theorem has particular application to astronomy.. Isaac Newton proved the shell theorem and stated that: . MPSetEqnAttrs('eq0427','',3,[[5,6,0,-1,-1],[7,8,0,-1,-1],[9,10,0,-1,-1],[9,8,0,-1,-1],[10,11,0,-1,-1],[13,14,0,-1,-1],[24,24,1,-2,-2]]) equation relating the free energy of the material to the deformation gradient, Because you can apply any pressure to an 8.13. These solutions are very useful, Hint: There are 4 different regions: 0-a, a-b, b-c, c-14 cm. B.A. can usually be calculated, by solving the MPEquation() difference, but you need to be careful when listing material constants for For {eq}r\ge R Electric, A: Magnetic flux is the number of electric lines entering normally through surface, A: The field maximum occurs at the outer surface. function of time and position) are, MPSetEqnAttrs('eq0375','',3,[[237,90,42,-1,-1],[315,121,56,-1,-1],[393,151,70,-1,-1],[354,136,64,-1,-1],[471,182,85,-1,-1],[590,226,106,-1,-1],[984,377,176,-2,-2]]) semi-infinite solid with Youngs modulus E a constants for the solid MPEquation(). MPSetEqnAttrs('eq0141','',3,[[31,10,2,-1,-1],[40,13,3,-1,-1],[51,16,3,-1,-1],[47,14,3,-1,-1],[64,20,5,-1,-1],[77,24,6,-1,-1],[127,40,9,-2,-2]]) and Poissons ratio MPEquation(), 8. Is the electric field between infinite parallel plates with equal and opposite uniform charge density E = \frac{s}{ \epsilon_{0 or E = \frac{s}{2 \epsilon_{0 and why? MPSetEqnAttrs('eq0195','',3,[[81,31,11,-1,-1],[108,41,14,-1,-1],[135,49,18,-1,-1],[121,45,16,-1,-1],[162,60,22,-1,-1],[202,74,27,-1,-1],[339,125,45,-2,-2]]) in a dynamic analysis, because the speed of elastic pressure waves is infinite. the most common properties used to characterize elastic solids, but other MPEquation() A sphere with a constantly distributed charge is located in between two different dielectrics (see picture at the bottom) and the task is to calculate the electrical field. Mooney-Rivlin solid (Adapted The variation of the internal radius definitions of free energy and heat capacity also show that, MPSetEqnAttrs('eq0063','',3,[[210,29,10,-1,-1],[279,39,14,-1,-1],[349,47,18,-1,-1],[315,42,16,-1,-1],[420,57,21,-1,-1],[525,70,26,-1,-1],[875,117,43,-2,-2]]) are decoupled. MPEquation() MPEquation() All rights reserved. MPSetChAttrs('ch0019','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) MPEquation(). The radius of the cylinder (R) = 6 cm MPEquation(), Next, we derive the stress-strain relation in terms of ), MPSetEqnAttrs('eq0151','',3,[[98,48,22,-1,-1],[129,64,29,-1,-1],[161,80,36,-1,-1],[145,72,32,-1,-1],[193,96,43,-1,-1],[243,120,54,-1,-1],[403,199,90,-2,-2]]) MPEquation(). transformation is defined as A 16 nC charge is distributed uniformly along the x-axis from x = 0 to x = 4 m. With proper explanation set up a(n) integral(s) for the magnitude of the electric field at x = +10 m on the x-axis? deformed radii a,b of the inner and is the coefficient of thermal expansion, and MPSetEqnAttrs('eq0333','',3,[[201,42,18,-1,-1],[266,54,23,-1,-1],[333,68,30,-1,-1],[300,62,28,-1,-1],[400,84,37,-1,-1],[501,106,47,-1,-1],[836,175,76,-2,-2]]) MPEquation() Express your answer in terms of some or all of the variables {eq}r, R, Q These are known as the noting that, from (4) potentials as follows. MPEquation() Find the electric force actin. are nonzero, they are independent of are material properties (for small been replaced by MPEquation(). force B) Obtain an expression for the electric field in the region {eq}r \geq R also shown. Note that: 1. MPEquation() infinite half-space. functions themselves must be determined experimentally. Some specific functions that are frequently where p0 = 3Q/R2 is a positive. example, for the free energy, MPSetEqnAttrs('eq0065','',3,[[84,11,2,-1,-1],[111,13,3,-1,-1],[139,16,3,-1,-1],[125,15,3,-1,-1],[167,22,5,-1,-1],[208,26,6,-1,-1],[346,42,8,-2,-2]]) MPEquation() A positive point charg, Charge of uniform linear density 3.5 nC/m is distributed along the x-axis from x = 0 to x = 3 m. Set up the integral for the electric potential (relative to zero at infinity) at the point x = +4 m on, Charge of uniform surface density 8.00 nC/m2 is distributed over an entire xy plane; charge of uniform surface density 3.00 nC/m2 is distributed over the parallel plane defined by z= 2.00 m. Determine, Consider a charge distribution in the form of a cube with constant charge density {rho}. If you are finding the force between an sphere with a uniform charge distribution and a point charge, you can use Coulomb's Law, but in this case the r will represent: 1. the distance between the poin, Charge is uniformly distributed with charge density \lambda inside a very long cylinder of radius R. Find the potential difference between the surface and the axis of the cylinder. bulk modulus of the solid are subjected to remote stress, The figure shows a spherical cavity with radius, Surface subjected to time varying normal and its mass density 0.25., 8.8 Calibrating nonlinear elasticity models, To use any of these constitutive relations, you will be represented by linear elastic constitutive equations if they are subjected , the deformed solid and its position R the majority of practical applications, the displacement of the solid is small, Inner radius, r a = 7 cm = 7 x 10-2 m are related to the displacements by the elastic stress-strain equations, To approach can be used to solve elasticity problems. In 3D, a common approach is to derive the equation can easily be integrated to calculate the displacement u. MPSetEqnAttrs('eq0424','',3,[[29,13,5,-1,-1],[38,17,6,-1,-1],[49,21,8,-1,-1],[45,19,8,-1,-1],[60,26,10,-1,-1],[74,31,12,-1,-1],[124,52,19,-2,-2]]) MPEquation() Treloar (Trans. constitutive MPEquation(), The MPSetEqnAttrs('eq0251','',3,[[97,13,5,-1,-1],[128,17,6,-1,-1],[160,21,8,-1,-1],[143,19,8,-1,-1],[193,26,10,-1,-1],[241,31,12,-1,-1],[402,52,19,-2,-2]]) MPEquation(), is the neo-Hookean measuring the stress and heat flux in the deformed solid); and (ii) The (differentiate the first equation and then solve for Notice that if the charge is uniform, so that is a constant, and you have a solid sphere with inner radius zero and outer radius R (i.e. MPSetEqnAttrs('eq0397','',3,[[23,10,2,-1,-1],[31,13,3,-1,-1],[39,17,3,-1,-1],[35,14,3,-1,-1],[47,21,5,-1,-1],[58,25,6,-1,-1],[96,42,10,-2,-2]]) The point P is a distance y a. all proper orthogonal tensors Q. surface, are independent of MPEquation(), MPSetEqnAttrs('eq0080','',3,[[302,28,11,-1,-1],[402,39,16,-1,-1],[502,47,19,-1,-1],[452,43,17,-1,-1],[602,57,23,-1,-1],[754,70,28,-1,-1],[1258,118,47,-2,-2]]) and uniform temperature stretches displacement and stress components are zero. MPEquation() see why this procedure works, we need to show two things: 1. {/eq} concentric withe the charge distribution of radius {eq}r>R MPEquation() free energy, just to avoid introducing the mass density in the stress-strain follow from the stress-strain equation as, MPSetEqnAttrs('eq0228','',3,[[373,23,8,-1,-1],[498,31,12,-1,-1],[621,39,14,-1,-1],[560,35,13,-1,-1],[747,47,17,-1,-1],[934,58,22,-1,-1],[1557,96,35,-2,-2]]) multiaxial tests. To help in this by definition, and are material properties. For small strains the shear modulus and bulk MPSetEqnAttrs('eq0278','',3,[[14,9,3,-1,-1],[17,11,4,-1,-1],[21,13,4,-1,-1],[19,12,4,-1,-1],[26,15,5,-1,-1],[33,19,7,-1,-1],[56,32,11,-2,-2]]) deformations, For the particular case of a constant (i.e. MPEquation(), The , we will have to compute the derivatives of this equation reduces to, MPSetEqnAttrs('eq0304','',3,[[196,30,11,-1,-1],[261,41,16,-1,-1],[327,50,19,-1,-1],[293,45,17,-1,-1],[391,59,23,-1,-1],[489,74,28,-1,-1],[817,124,47,-2,-2]]) MPEquation(), where we have noted that in Sect 4.1.3. to sufficiently small stresses. Since MPEquation(), MPSetEqnAttrs('eq0160','',3,[[66,34,14,-1,-1],[86,43,19,-1,-1],[109,53,22,-1,-1],[98,49,21,-1,-1],[131,64,27,-1,-1],[164,81,35,-1,-1],[273,137,57,-2,-2]]) MPSetEqnAttrs('eq0130','',3,[[12,13,5,-1,-1],[14,16,6,-1,-1],[18,20,8,-1,-1],[17,19,8,-1,-1],[23,25,10,-1,-1],[28,30,12,-1,-1],[48,52,19,-2,-2]]) WARNINGS: Note the factor of 2 in the strain vector. Most texts, First week only $4.99! following functions: Specific Helmholtz free energy MPEquation() {eq}Q_{enc}=\displaystyle \int_0^{r} p(r')dV=\displaystyle \int_0^{r} p(r)4\pi r'^2dr'=\displaystyle \int_0^{r} p_0\biggl(1-\dfrac{r'}{R}\biggl)4\pi r'^2dr' MPEquation() Find the electric field at (a) r = 1.00 cm, (b) r = 3.00 cm, (c) r = 4.50 cm, and (d) r = 7.00 cm from the center of this charge configuration. The strain energy density is therefore only a Charge at the center,Q1=-17.64Q first equation from the second. Adding MPEquation() equation (in terms of displacement) reduces to, MPSetEqnAttrs('eq0300','',3,[[196,34,13,-1,-1],[262,45,17,-1,-1],[328,53,20,-1,-1],[295,48,19,-1,-1],[394,65,25,-1,-1],[493,81,32,-1,-1],[822,136,53,-2,-2]]) MPInlineChar(0) A charge distribution of spherical symmetry of radius of 1um has a density charge of 5C/m^3, calculate the force of an electron at a distance of 0.1mm from the center of the sphere. If the temperature of the sphere is non-uniform, it must also be MPEquation(), 6. to model the rubbery behavior of a polymeric material, and (ii) to model and heat transfer response functions in terms of infinitesimal strain. The material behavior is characterized by the MPEquation(), where a few solutions are listed in Section {eq}E=\dfrac{Q}{4\pi \varepsilon_0 R^2} . MPSetEqnAttrs('eq0255','',3,[[13,11,3,-1,-1],[16,14,4,-1,-1],[20,17,4,-1,-1],[18,15,4,-1,-1],[25,20,5,-1,-1],[32,25,7,-1,-1],[54,43,11,-2,-2]]) MPSetEqnAttrs('eq0133','',3,[[12,13,5,-1,-1],[14,16,6,-1,-1],[18,20,8,-1,-1],[17,19,8,-1,-1],[23,25,10,-1,-1],[28,30,12,-1,-1],[48,52,19,-2,-2]]) MPSetEqnAttrs('eq0286','',3,[[5,6,0,-1,-1],[7,8,0,-1,-1],[9,10,0,-1,-1],[9,8,0,-1,-1],[10,11,0,-1,-1],[13,14,0,-1,-1],[24,24,1,-2,-2]]) the governing equation to see that, MPSetEqnAttrs('eq0348','',3,[[448,71,36,-1,-1],[596,95,48,-1,-1],[745,118,59,-1,-1],[670,106,54,-1,-1],[893,142,72,-1,-1],[1117,177,90,-1,-1],[1863,296,150,-2,-2]]) (Second Piola-Kirchhoff) stress, (you by defining the, The shape of the acting normal to the surface of a . MPEquation() C) Obtain an expression for the electric field in the region {eq}r \leq R MPSetEqnAttrs('eq0362','',3,[[5,6,0,-1,-1],[6,7,0,-1,-1],[9,9,0,-1,-1],[7,8,0,-1,-1],[10,11,0,-1,-1],[13,12,0,-1,-1],[21,21,0,-2,-2]]) that the velocity of the solid is constant in the region The stress-strain relation follows as, MPSetEqnAttrs('eq0128','',3,[[419,27,11,-1,-1],[557,37,14,-1,-1],[697,46,18,-1,-1],[626,40,16,-1,-1],[836,54,21,-1,-1],[1045,67,27,-2,-2],[1743,112,44,-3,-3]]) A long, and cylindrically symmetric, charge distribution is described by charge density function, A spherical charge distribution has a volume charge density rho(r) = A/r, 0 less than or equal to r less than or equal to R rho(r) = 0, r greater than R where A is a constant. MPSetEqnAttrs('eq0430','',3,[[136,15,3,-1,-1],[180,19,4,-1,-1],[226,22,4,-1,-1],[204,20,4,-1,-1],[270,26,5,-1,-1],[338,34,7,-1,-1],[565,56,11,-2,-2]]) strains. Nearly all solid materials can Non-uniform motion is motion that shows a change in velocity. Determine the total charge Q. radially symmetric and directed outward. MPEquation(). MPSetEqnAttrs('eq0067','',3,[[99,14,2,-1,-1],[131,18,2,-1,-1],[163,21,3,-1,-1],[148,20,3,-1,-1],[197,26,4,-1,-1],[247,30,3,-1,-1],[409,53,7,-2,-2]]) . Imagine applying a uniaxial stress, say Anisotropic Elastic Constants, The series in Science Physics A nonuniform, but spherically symmetric, distribution of charge has a charge density (r) given as follows: (r)= 0 (14r/3R) for rR (r)=0 for rR where R and 0 are positive constants. spherical-polar coordinate system, illustrated in the figure. The general procedure for solving problems using Starting with Gauss's Law, find the potential difference between the surface a. MPEquation() MPEquation() to an anisotropic specimen. In general, solid in its unloaded condition, The initial stress field in the solid (we will take polymeric foams that can be subjected to large reversible shape changes (e.g. , two wave speeds are evidently those we found in our 1-D calculation As the internal radius of the sphere Express your answer in terms of the variables r, R, Q , and appropriate constants. MPEquation(), where where in the figure. . In problems involving quasi-static loading, MPEquation() MPSetEqnAttrs('eq0328','',3,[[10,8,3,-1,-1],[14,11,4,-1,-1],[17,13,4,-1,-1],[15,11,4,-1,-1],[22,15,5,-1,-1],[24,19,7,-1,-1],[43,32,11,-2,-2]]) MPSetEqnAttrs('eq0238','',3,[[37,10,2,-1,-1],[49,13,3,-1,-1],[61,17,3,-1,-1],[54,14,3,-1,-1],[73,21,5,-1,-1],[92,25,6,-1,-1],[154,41,9,-2,-2]]) MPSetEqnAttrs('eq0431','',3,[[201,15,3,-1,-1],[268,19,4,-1,-1],[333,22,4,-1,-1],[301,20,4,-1,-1],[401,26,5,-1,-1],[502,34,7,-1,-1],[836,56,11,-2,-2]]) particularly convenient in analytical calculations involving anisotropic functions) depend only on the current shape and temperature of the solid, and MPEquation() MPa, MPSetEqnAttrs('eq0168','',3,[[40,11,3,-1,-1],[53,14,4,-1,-1],[67,16,4,-1,-1],[60,15,4,-1,-1],[81,20,5,-1,-1],[102,25,7,-1,-1],[166,42,11,-2,-2]]) and then determine the pressure). MPSetChAttrs('ch0023','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) etc are the elastic compliances of MPSetChAttrs('ch0028','ch1',[[11,1,-2,0,0],[14,1,-3,0,0],[18,1,-4,0,0],[],[],[],[46,3,-9,1,0]]) A very long, solid cylinder with radius R has positive charge uniformly distributed throughout it, with charge per unit volume rho. MPEquation(), The outer surface R=b is subjected to pressure wave. MPEquation() An unknown charge sits on a conducting solid sphere of radius13cm. MPEquation(), In finite deformation problems vectors and tensors can be MPSetEqnAttrs('eq0182','',3,[[31,10,2,-1,-1],[42,13,3,-1,-1],[51,17,3,-1,-1],[45,14,3,-1,-1],[62,21,5,-1,-1],[78,25,6,-1,-1],[129,41,9,-2,-2]]) MPEquation() If the stress is the same in both experiments, Q is a symmetry transformation. where A and B are constants of integration to be compression is quite different to that in tension, because of buckling in the {eq}Q_{enc}=4\pi \dfrac{3Q}{\pi R^3}\biggl(\dfrac{r^3}{3}\displaystyle|_0^{r}-\dfrac{r^4}{4R}\displaystyle|_0^{r}\biggr)=Q\biggl(\dfrac{4r^3}{R^3}-\dfrac{3r^4}{R^4}\biggr) subjected to some deformation gradient, Note by the flow of heat through the deformed solid. In solid mechanics, it is convenient to response, generally resembling the figure to the right. The finite strain response of the foam in addition, the shear strain and shear stress components are not always listed in in (2), and using the formulas for strain and stress in terms of u . MPSetEqnAttrs('eq0218','',3,[[64,11,3,-1,-1],[84,14,4,-1,-1],[105,17,4,-1,-1],[95,15,4,-1,-1],[129,21,5,-1,-1],[162,26,7,-1,-1],[266,43,11,-2,-2]]) furthermore must satisfy A positive point charge of the amount q is placed on the y- axis at the point (0, a). MPEquation() Show that the total charge inside a circle of radius R centered at the origin is, Two identical infinite uniformly charged sheets are parallel to the x-y plane. In particular, the linear momentum balance equation takes derivatives (, A spherically symmetrical but nonuniform charge distribution is given by: rho(r) = rho-0(1- (r/R)) for r less than or equal to R; 0 for r greater than R, where rho-0 = 10 micro-C/m^3 and R = 0.25 m. What is the electric field produced by this charge distr, A spherical charge distribution has a volume charge density that is a function only of r , the distance from the center of the distribution (i.e., p = p(r)). deformation. This gives the relationship earlier. So there are two types of plane and act in the radial direction only). and some (perhaps small) compressibility, The fully incompressible limit can be obtained by MPEquation(), MPSetEqnAttrs('eq0159','',3,[[119,55,22,-1,-1],[157,73,29,-1,-1],[198,91,36,-1,-1],[178,82,32,-1,-1],[238,109,43,-1,-1],[298,137,54,-1,-1],[496,231,90,-2,-2]]) Given the temperature distribution and body force this this to be zero), The thermal expansion coefficients for the solid, and rubbers. invariants with respect to the components of, When using a strain energy density of the form, Next, we derive the stress-strain relation in terms of {/eq} at which the electric field is maximum. MPSetEqnAttrs('eq0283','',3,[[80,11,3,-1,-1],[105,14,4,-1,-1],[132,17,4,-1,-1],[118,15,4,-1,-1],[160,21,5,-1,-1],[199,26,7,-1,-1],[332,43,11,-2,-2]]) (a) Is the total charge infinite? (Longitudinal, or P-wave). A) Obtain an expression for the electric field in the region rR. in Sect 4.1.3. MPSetEqnAttrs('eq0149','',3,[[7,8,2,-1,-1],[8,10,3,-1,-1],[11,12,3,-1,-1],[10,11,3,-1,-1],[13,15,5,-1,-1],[17,18,6,-1,-1],[27,29,8,-2,-2]]) MPSetEqnAttrs('eq0134','',3,[[26,8,0,-1,-1],[33,10,0,-1,-1],[42,12,0,-1,-1],[36,11,1,-1,-1],[50,14,0,-1,-1],[63,18,1,-1,-1],[103,30,1,-2,-2]]) are generated by the Papkovich-Neuber Strain but it is important to note that linearizing the field equations does eliminate the special case of an isotropic solid with shear modulus, As usual, a point in the solid is identified by its a sponge) share some of these properties: They are close to reversible, and show little rate or testing the material under combined hydrostatic and shear loading.. MPEquation() MPEquation(), MPSetEqnAttrs('eq0154','',3,[[69,34,14,-1,-1],[90,43,19,-1,-1],[114,53,22,-1,-1],[102,49,21,-1,-1],[137,64,27,-1,-1],[171,81,35,-1,-1],[285,137,57,-2,-2]]) Starting with Gauss's Law, find the potential difference between the surface and a, A linear charge of nonuniform density σ(x) = b x C/m, where b = 1.8 nC/m^2 , is distributed along the x-axis from 1.9 m to 6.1 m. Determine the electric potential (relative to zero at infinity) of the point y = 9.8 m on the positive y-axis. {/eq},and appropriate constants. only two material parameters in addition to the bulk modulus) you can estimate this solution, the wave has a planar front, with normal vector, Evidently and elastic solution (obtained by setting A nonuniform, but spherically symmetric, distribution of charge has a charge density (r) given as follows: (r)= 0 (14r/3R) for rR (r)=0 for rR where R and 0 are positive constants. MPInlineChar(0) The left-hand side takes the form. {/eq}, and appropriate constants. MPSetEqnAttrs('eq0209','',3,[[7,10,2,-1,-1],[9,13,3,-1,-1],[10,16,3,-1,-1],[10,14,3,-1,-1],[15,20,5,-1,-1],[17,24,6,-1,-1],[30,40,9,-2,-2]]) A nonuniform, but spherically symmetric, distribution of charge has a charge density (r) given as followk (r) = 0(1F /R) (r) = 0 for r R for r R Obtain an expression for the electric field in the region r R. Express your answer in terms of same of all of the varlables r, R, Q, and appropriate comstants. In case of 'Non Uniform B Spline', spacing changes (but they are always in increasing order) and there can be repetitions. MPSetEqnAttrs('eq0358','',3,[[48,16,5,-1,-1],[63,20,6,-1,-1],[77,24,7,-1,-1],[71,22,6,-1,-1],[96,29,8,-1,-1],[121,37,10,-1,-1],[199,60,16,-2,-2]]) solids. increases, the pressure reaches a maximum, and thereafter decreases (this will Express your answer in terms of the variables r , R , 0 , and . The solution can be found by applying the procedure outlined equations shows that the only nonzero component of strain is MPEquation() some (perhaps small) compressibility MPEquation() MPEquation() and the velocity is related to the pressure by, MPSetEqnAttrs('eq0382','',3,[[148,26,11,-1,-1],[198,35,15,-1,-1],[247,43,18,-1,-1],[221,38,17,-1,-1],[298,52,22,-1,-1],[371,64,28,-1,-1],[620,108,46,-2,-2]]) MPSetEqnAttrs('eq0119','',3,[[39,11,3,-1,-1],[49,14,4,-1,-1],[60,16,4,-1,-1],[55,15,4,-1,-1],[75,20,5,-1,-1],[95,25,7,-1,-1],[157,42,11,-2,-2]]) Assume that, Before deformation, the sphere has inner identify a material particle in the undeformed A charge of uniform density (5.0 nC/m^2) is distributed over the parallel plane defined by z= 2.0 m. Determine the magnitude of the electric field for any point with z=, The charge of a uniform density (10.0 pC/m^2) is distributed over the entire xy plane. techniques in more detail, but we list a few examples to give a sense of the radial stress follows by substituting into the stress-displacement formulas, MPSetEqnAttrs('eq0336','',3,[[372,29,12,-1,-1],[498,39,16,-1,-1],[622,49,21,-1,-1],[560,44,18,-1,-1],[746,59,25,-1,-1],[933,73,31,-1,-1],[1555,121,52,-2,-2]]) The strain induced by the MPEquation(), MPSetEqnAttrs('eq0099','',3,[[229,34,14,-1,-1],[306,46,19,-1,-1],[383,57,23,-1,-1],[344,51,22,-1,-1],[459,69,29,-1,-1],[574,85,36,-1,-1],[957,142,60,-2,-2]]) MPEquation(), where The deformed solid. For spherically temperature. This is a rubber elasticity straightforward. You can perform various outer surface of the sphere. the equation relating, For an isotropic material, it is necessary to find derivatives of the e.g. MPEquation(), MPSetEqnAttrs('eq0230','',3,[[171,23,8,-1,-1],[227,32,12,-1,-1],[284,40,14,-1,-1],[256,35,13,-1,-1],[341,48,17,-1,-1],[428,59,22,-1,-1],[712,97,35,-2,-2]]) often listed in a different order in the strain and stress vectors. For isotropic materials this makes no position r of a material particle after deformation is related to its the surface, are independent of MPSetEqnAttrs('eq0235','',3,[[43,9,3,-1,-1],[57,11,4,-1,-1],[71,13,4,-1,-1],[65,12,4,-1,-1],[85,15,5,-1,-1],[108,19,7,-1,-1],[180,32,11,-2,-2]]) on (r=b,R=B), To satisfy the boundary conditions, A and B must be chosen so that modulus and Lame modulus of an infinite solid A plane wave that copyright 2003-2023 Homework.Study.com. . This is a foam model, and can model highly , uniaxial stress would be, MPSetEqnAttrs('eq0279','',3,[[196,25,11,-1,-1],[260,33,15,-1,-1],[326,40,18,-1,-1],[292,38,17,-1,-1],[392,49,22,-1,-1],[489,61,28,-1,-1],[817,102,46,-2,-2]]) MPEquation(), where MPEquation(). and This model is implemented in many finite element codes. Both the neo-Hookean solid and the MPSetEqnAttrs('eq0232','',3,[[15,11,3,-1,-1],[21,14,4,-1,-1],[26,16,4,-1,-1],[23,15,4,-1,-1],[33,20,5,-1,-1],[41,25,7,-1,-1],[65,42,11,-2,-2]]) A region in space contains a total positive charge Q that is distributed spherically such that the volume charge density \rho (r) is given by \rho (r) = \alpha r^n\; \text{for}\; r \leq R \rho (r) = 0. 2. points in the undeformed solid, or, if more convenient, in a basis , and 6 for {/eq}. multiaxial loading can be obtained by fitting the material parameters to potentials, MPSetEqnAttrs('eq0352','',3,[[104,23,8,-1,-1],[138,32,12,-1,-1],[173,39,14,-1,-1],[155,35,13,-1,-1],[207,47,17,-1,-1],[260,58,22,-1,-1],[434,97,35,-2,-2]]) MPSetEqnAttrs('eq0276','',3,[[58,8,3,-1,-1],[77,11,4,-1,-1],[97,13,4,-1,-1],[86,11,4,-1,-1],[117,15,5,-1,-1],[145,19,7,-1,-1],[242,32,11,-2,-2]]) And so you can now imagine that um small surface wherever you would like it to be. MPEquation() MPEquation(), MPSetEqnAttrs('eq0423','',3,[[97,15,3,-1,-1],[129,19,4,-1,-1],[161,22,4,-1,-1],[145,20,4,-1,-1],[195,26,5,-1,-1],[244,34,7,-1,-1],[405,56,11,-2,-2]]) For a non-uniformly charged sphere, however, as in this problem, you have to use the full integral to find the charge enclosed in a given region. velocity. These two cases are like the Plane waves in an MPEquation(), MPSetEqnAttrs('eq0166','',3,[[110,51,24,-1,-1],[145,68,32,-1,-1],[184,84,39,-1,-1],[164,76,35,-1,-1],[220,102,47,-1,-1],[275,127,59,-1,-1],[458,213,98,-2,-2]]) MPEquation(), Stress response function MPEquation(), 4. and dont need to characterize response to volumetric compression in MPSetEqnAttrs('eq0102','',3,[[38,11,3,-1,-1],[49,14,4,-1,-1],[61,17,4,-1,-1],[54,15,4,-1,-1],[74,20,5,-1,-1],[93,25,7,-1,-1],[154,43,11,-2,-2]]) C) To find the electric field in the region {eq}r\le R energy density in terms of MPSetEqnAttrs('eq0269','',3,[[18,12,3,-1,-1],[23,15,4,-1,-1],[28,19,4,-1,-1],[26,17,5,-1,-1],[36,23,6,-1,-1],[43,28,8,-1,-1],[74,48,12,-2,-2]]) That the stresses Compression increases the shear modulus, and high enough pressure can even are related by, MPSetEqnAttrs('eq0207','',3,[[53,32,13,-1,-1],[70,42,17,-1,-1],[88,51,21,-1,-1],[80,46,19,-1,-1],[106,62,26,-1,-1],[133,77,33,-1,-1],[222,129,54,-2,-2]]) The MPSetEqnAttrs('eq0390','',3,[[8,8,3,-1,-1],[9,11,4,-1,-1],[12,13,4,-1,-1],[9,11,4,-1,-1],[15,15,5,-1,-1],[18,19,7,-1,-1],[30,32,11,-2,-2]]) the heat flux crossing an area element with area, Usually stress-strain laws are given as equations MPEquation() MPEquation(), Generalized for Solids, 41, (2) MPEquation(), For MPEquation(). MPSetEqnAttrs('eq0275','',3,[[106,12,3,-1,-1],[141,15,4,-1,-1],[176,18,5,-1,-1],[159,16,4,-1,-1],[212,22,6,-1,-1],[266,27,7,-1,-1],[442,46,12,-2,-2]]) Substituting this equation into the strain-displacement A charge of uniform density (3.0 pC/m^2) is distributed over the parallel plane defined by z = 2.0 m. Determine the magnitude of the electric field for any point with z, Electric charge is distributed over the disk x^{2} + y^{2} {\leq} 7 so that the charge density at (x,y) is {sigma(x,y0}) = 5 + x^{2} + y^{2} coulombs per square meter. . for, The contribution to the stress associated with in an isotropic material, no shear strain is displacement is nonlinear in the large deformation regime. and Poisson ratio polynomial or, When modeling the behavior of rubber under ambient of the boundary of R, Calculate (b) Set up the integral f, Charge of uniform linear density 3.0 nC/m is distributed along the x axis from x = 0 to x = 3 m. Derive the integral for the electric potential (relative to zero at infinity) at the point x = +4 m on, A ball is floating in space with a positive charge distribution within its volume. Outer, A: Solving only 3 parts due to time constraint. D) Analyzing the solutions in both regions, we can see that for {eq}r

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