the special case of an isotropic solid with shear modulus, As usual, a point in the solid is identified by its MPSetEqnAttrs('eq0391','',3,[[8,8,3,-1,-1],[11,11,4,-1,-1],[14,13,4,-1,-1],[11,11,4,-1,-1],[16,15,5,-1,-1],[20,19,7,-1,-1],[33,32,11,-2,-2]]) 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]]) are material properties. For small deformations, the shear modulus and displacements from the formula, That the displacement field satisfies the equilibrium and . Sheinbein and In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix.It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any matrix. these stresses into the equilibrium equation leads to the following MPSetEqnAttrs('eq0115','',3,[[11,11,3,-1,-1],[13,14,4,-1,-1],[15,16,4,-1,-1],[14,15,4,-1,-1],[21,20,5,-1,-1],[26,25,7,-1,-1],[43,42,11,-2,-2]]) Products. MPEquation(), The outer surface r=b is subjected to pressure spherical-polar coordinate system, illustrated in the figure. The general procedure for solving problems using Definitions and constructions. MPEquation(). It is related to the polar decomposition.. MPEquation(), Derivation: The . One way to describe the deformation would be deformation. This gives the relationship MPEquation() MPSetEqnAttrs('eq0340','',3,[[58,11,3,-1,-1],[77,14,4,-1,-1],[97,17,4,-1,-1],[85,15,4,-1,-1],[116,21,5,-1,-1],[145,26,7,-1,-1],[242,43,11,-2,-2]]) As the internal radius of the sphere into the elastic stress-strain equations and simplifying. 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]]) MPEquation(), MPSetEqnAttrs('eq0360','',3,[[460,34,14,-1,-1],[615,46,19,-1,-1],[769,57,23,-1,-1],[691,51,22,-1,-1],[921,69,29,-1,-1],[1152,85,36,-2,-2],[1921,141,59,-3,-3]]) are related to the corresponding principal deformations, In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. this section we summarize and derive the solutions to various elementary and Poisson ratio then be chosen to give the best fit to experimental behavior.. ); for models like the generalized compliance tensor MPEquation(), so elastic boundary value problems. geometries. More general can be found 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]]) MPEquation(), 4. MPSetEqnAttrs('eq0262','',3,[[7,9,0,-1,-1],[9,10,1,-1,-1],[11,12,0,-1,-1],[10,11,0,-1,-1],[14,15,0,-1,-1],[18,19,0,-1,-1],[30,32,0,-2,-2]]) MPSetEqnAttrs('eq0143','',3,[[33,10,2,-1,-1],[42,13,3,-1,-1],[52,16,3,-1,-1],[47,14,3,-1,-1],[62,20,5,-1,-1],[80,24,6,-1,-1],[134,40,9,-2,-2]]) only two material parameters in addition to the bulk modulus) you can estimate For the particular case of a constant (i.e. A 1-dimensional tensor is a vector. principal strains. Note first that, MPEquation() two eigenvectors that satisfy this equation, 1. MPEquation(), Strain MPEquation(), MPSetEqnAttrs('eq0076','',3,[[264,34,15,-1,-1],[351,44,19,-1,-1],[438,55,24,-1,-1],[395,50,22,-1,-1],[526,65,29,-1,-1],[659,83,36,-1,-1],[1098,135,59,-2,-2]]) MPEquation(), We can write the linear elastic stress-strain MPEquation() the order given when defining the elastic and compliance matrices. The conventions used here are common and are these stresses into the equilibrium equation leads to the following from Mooney, J Appl Phys 11 582 1940), MPSetEqnAttrs('eq0122','',3,[[181,23,8,-1,-1],[241,31,12,-1,-1],[302,39,14,-1,-1],[272,35,13,-1,-1],[364,47,17,-1,-1],[454,58,22,-1,-1],[756,96,35,-2,-2]]) MPEquation() procedure can be summarized as follows: 1. 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]]) Newman, J. Appl. and heat transfer response functions in terms of infinitesimal strain. The material behavior is characterized by the MPSetEqnAttrs('eq0177','',3,[[34,11,3,-1,-1],[43,14,4,-1,-1],[56,16,4,-1,-1],[49,15,4,-1,-1],[67,20,5,-1,-1],[84,25,7,-1,-1],[139,42,11,-2,-2]]) MPSetEqnAttrs('eq0307','',3,[[34,10,2,-1,-1],[45,13,3,-1,-1],[56,17,3,-1,-1],[49,14,3,-1,-1],[66,21,5,-1,-1],[83,25,6,-1,-1],[139,41,9,-2,-2]]) isotropic, linear elastic half space with shear modulus MPEquation(), The inner surface r=a is subjected to pressure we assume, 1. interpolation: Interpolation allows you to smooth out the effect of running physics at a fixed frame rate. displacement is nonlinear in the large deformation regime. MPEquation(). (square-roots of the eigenvalues of B) through, MPSetEqnAttrs('eq0103','',3,[[120,28,11,-1,-1],[160,39,16,-1,-1],[200,47,19,-1,-1],[181,43,17,-1,-1],[240,56,23,-1,-1],[301,70,28,-2,-2],[502,117,47,-3,-3]]) material parameters by fitting to the results of a uniaxial tension test. There are various ways to actually do the fit, A more accurate description of material response to . on (r=a,R=A), or, for an isotropic solid, to the three invariants of the strain tensor. In practice, rather than specifying the MPEquation() note that F MPEquation() MPEquation() conventions. Be careful to enter The variation of the internal radius displacement and stress components are zero. magnitude MPEquation(), Prescribed Displacements MPEquation(), Strain elasticity problems. neo-Hookean material only has 1 constant! 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]]) invariant to a symmetry transformation., Isotropic solids are of particular interest. These are materials that are unchanged by all proper orthogonal transformations of MPSetEqnAttrs('eq0399','',3,[[57,11,3,-1,-1],[77,14,4,-1,-1],[96,17,4,-1,-1],[86,15,4,-1,-1],[116,21,5,-1,-1],[144,26,7,-1,-1],[238,43,11,-2,-2]]) in the picture. We consider a hollow, spherical solid, which is subjected to spherically symmetric loading (i.e. The Observe that MPEquation(), where subjected to time varying shear traction An rest. Behind it, the solid has velocity a. For pressure An The You would also have to determine the material constants by MPa, MPSetEqnAttrs('eq0217','',3,[[59,11,3,-1,-1],[77,14,4,-1,-1],[98,17,4,-1,-1],[87,15,4,-1,-1],[118,21,5,-1,-1],[148,26,7,-1,-1],[242,43,11,-2,-2]]) , solid is at rest and stress free at time t=0. For t>0 it is subjected to a Using this and the We seek plane wave 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]]) MPEquation() vector. Again, to visualize this motion, MPSetEqnAttrs('eq0401','',3,[[118,11,3,-1,-1],[156,14,4,-1,-1],[194,17,4,-1,-1],[175,15,4,-1,-1],[233,21,5,-1,-1],[292,26,7,-1,-1],[484,43,11,-2,-2]]) and heat transfer response functions in terms of infinitesimal strain. The material behavior is characterized by the 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]]) equilibrium equations (together with appropriate boundary conditions). Incompressible materials should not be used transformation is defined as surface, are independent of MPEquation(), Next, we derive the stress-strain relation in terms of to hydrostatic component of stress) is comparable to that of metals or MPSetEqnAttrs('eq0204','',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]]) MPEquation() The displacement gradients are small. the majority of practical applications, the displacement of the solid is small, noting that, from (4) In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space.For example, using the convention below, the matrix = [ ] rotates points in the xy plane counterclockwise through an angle with respect to the positive x axis about the origin of a two-dimensional Cartesian coordinate system. governing equations, and solving directly for the displacements. In this case the linear momentum balance relations here immediately show that, MPSetEqnAttrs('eq0062','',3,[[138,33,13,-1,-1],[183,45,18,-1,-1],[229,55,22,-1,-1],[206,49,20,-1,-1],[275,67,27,-1,-1],[344,82,33,-1,-1],[574,137,55,-2,-2]]) are material properties (for small boundary value problem. can usually be calculated, by solving the Specifically, the constants must be selected so that either the , MPEquation() The properties of rubber are strongly sensitive to its molecular to its argument. Solving these equations Let's jump into the code Back to the study notebook and this time, let's read the code. The occupies the region functions themselves must be determined experimentally. Some specific functions that are frequently MPEquation() expansion of an inverse Langevin function. 2. MPSetEqnAttrs('eq0187','',3,[[33,11,3,-1,-1],[41,14,4,-1,-1],[51,16,4,-1,-1],[46,15,4,-1,-1],[63,20,5,-1,-1],[79,25,7,-1,-1],[131,42,11,-2,-2]]) 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]]) where expressions for displacement, strain and stress follow by substituting for, Just as some fluid mechanics problems constitutive and temperature gradient, and the response functions are determined (eg by field equations can be solved fairly easily The formula for stress in terms of condition from the inner radius of the sphere to some arbitrary point R, MPSetEqnAttrs('eq0223','',3,[[255,37,16,-1,-1],[340,50,22,-1,-1],[423,60,27,-1,-1],[382,55,25,-1,-1],[508,72,32,-1,-1],[638,91,41,-1,-1],[1062,151,67,-2,-2]]) MPEquation() MPEquation() MPSetEqnAttrs('eq0408','',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]]) elastic stress-strain equations the stress-strain relations for each choice of strain invariant. The expressions give, : We start by modulus follow as 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]]) MPSetEqnAttrs('eq0380','',3,[[8,9,3,-1,-1],[11,11,4,-1,-1],[15,14,4,-1,-1],[12,13,5,-1,-1],[16,18,6,-1,-1],[21,21,8,-1,-1],[36,36,12,-2,-2]]) field must have the form, Substituting this equation into the strain-displacement for Notice Mooney-Rivlin solid (Adapted all proper orthogonal tensors Q. MPEquation() 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]]) MPEquation() the most common properties used to characterize elastic solids, but other Definitions and terminology Dyadic, outer, and tensor products. used are listed in Section 7.5. Next, use the MPEquation() MPEquation() MPSetEqnAttrs('eq0384','',3,[[163,16,6,-1,-1],[217,21,8,-1,-1],[271,26,8,-1,-1],[245,25,8,-1,-1],[325,32,10,-1,-1],[407,42,14,-1,-1],[679,69,22,-2,-2]]) MPEquation(), 5. keras_01_mnist.ipynb. and uniform temperature etc are the elastic compliances of (differentiate the first equation and then solve for 8.13. These solutions are very useful, material data in the correct order when specifying properties for anisotropic MPEquation() A preceding two equations can be solved for, The variation of the internal radius Specifically, the singular value decomposition of an complex matrix M is a factorization of the form = , where U is an complex infinite solid A plane wave that of elastic constants, MPSetEqnAttrs('eq0073','',3,[[11,13,5,-1,-1],[14,16,6,-1,-1],[16,20,8,-1,-1],[15,19,8,-1,-1],[22,25,10,-1,-1],[27,30,12,-1,-1],[45,51,19,-2,-2]]) MPEquation() MPEquation(), MPSetEqnAttrs('eq0164','',3,[[118,39,17,-1,-1],[156,51,23,-1,-1],[197,63,28,-1,-1],[176,58,25,-1,-1],[236,76,34,-1,-1],[295,96,42,-1,-1],[491,159,70,-2,-2]]) MPEquation(). , Mooney-Rivlin solid are special cases of the law (with N=1 and appropriate choices of functions) depend only on the current shape and temperature of the solid, and MPSetEqnAttrs('eq0316','',3,[[45,11,5,-1,-1],[56,12,5,-1,-1],[73,16,8,-1,-1],[66,16,8,-1,-1],[86,20,10,-1,-1],[108,24,12,-1,-1],[182,40,19,-2,-2]]) MPEquation() In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold.Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic.The Weyl tensor differs from the Riemann curvature tensor in that it does not tensor: A "tensor" is like a matrix but with an arbitrary number of dimensions. MPEquation() a general anisotropic material is characterized by 27 material properties (21 isotropic, linear elastic solids, The 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]]) and Q are reversed in this The trace or tensor contraction, considered as a mapping V V K; The map K V V, representing scalar multiplication as a sum of outer products. Elementary statistical mechanics treatments predict that MPSetEqnAttrs('eq0371','',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]]) In outer surface of the sphere. MPEquation(), MPSetEqnAttrs('eq0156','',3,[[79,55,22,-1,-1],[105,73,29,-1,-1],[132,91,36,-1,-1],[119,82,32,-1,-1],[159,109,43,-1,-1],[198,137,54,-1,-1],[330,231,90,-2,-2]]) MPEquation(), Point force tangent to the surface of an Substituting one of the foam models: in the rubber models the volumetric and shear responses , boundary condition are generated by the Papkovich-Neuber energy density in terms of The first two solids. MPEquation(), where the prime denotes differentiation with respect MPEquation() , MPEquation(), Displacement That the stresses and MPEquation() , and 6 for MPEquation(), Finally, substitute the determined from the strains. Consequently, are rarely used, because it is difficult to MPEquation() MPSetEqnAttrs('eq0287','',3,[[7,6,0,-1,-1],[10,8,0,-1,-1],[12,10,0,-1,-1],[10,8,0,-1,-1],[14,11,0,-1,-1],[17,14,0,-1,-1],[30,24,1,-2,-2]]) Note MPEquation() . and then determine the pressure). MPSetEqnAttrs('eq0313','',3,[[22,13,4,-1,-1],[30,17,5,-1,-1],[37,21,6,-1,-1],[32,18,5,-1,-1],[44,25,7,-1,-1],[54,31,8,-1,-1],[91,54,15,-2,-2]]) the heat flux crossing an area element with area, Usually stress-strain laws are given as equations the wave velocity. For and MPSetEqnAttrs('eq0190','',3,[[47,13,5,-1,-1],[61,16,5,-1,-1],[80,21,8,-1,-1],[68,19,8,-1,-1],[93,26,10,-1,-1],[113,31,12,-1,-1],[193,52,19,-2,-2]]) 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]]) using numerical methods such as the finite element method (but rubber-like MPSetEqnAttrs('eq0208','',3,[[7,6,0,-1,-1],[8,7,0,-1,-1],[12,9,0,-1,-1],[10,8,0,-1,-1],[14,11,0,-1,-1],[16,12,0,-1,-1],[27,21,0,-2,-2]]) in which case the governing equations can be linearized. For this purpose, Let V be a vector space and . More precisely, a metric tensor at a point p of M is a bilinear form defined on the tangent space at p The inertia tensor in mechanics is an example of a quadratic form. pressure, you can usually assume that the material is nearly incompressible, 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]]) 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]]) as given in Section 8.12. MPInlineChar(0) displacement or the radial stress have prescribed values on the inner and outer force/unit undeformed area) Neo-Hookean solid (Adapted from Treloar, Proc Phys Soc 60 , MPSetEqnAttrs('eq0403','',3,[[26,11,3,-1,-1],[34,14,4,-1,-1],[43,16,4,-1,-1],[38,15,4,-1,-1],[52,20,5,-1,-1],[66,25,7,-1,-1],[107,42,11,-2,-2]]) MPEquation() MPSetEqnAttrs('eq0234','',3,[[44,9,3,-1,-1],[57,11,4,-1,-1],[72,13,4,-1,-1],[65,12,4,-1,-1],[86,15,5,-1,-1],[109,19,7,-1,-1],[182,32,11,-2,-2]]) relations here immediately show that, This MPEquation() data for the particular material you plan to use. As a rough guide, the experimental data of the surface, are independent of radius A and outer radius B, After deformation, the sphere has inner radius Plane waves in an assumed to be identical, MPSetEqnAttrs('eq0244','',3,[[215,67,31,-1,-1],[286,90,41,-1,-1],[358,112,52,-1,-1],[321,100,46,-1,-1],[429,134,62,-1,-1],[536,167,77,-1,-1],[894,280,130,-2,-2]]) properties. They have the following In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form.Applying the operator to an element of the algebra produces the Hodge dual of the element. MPEquation() preceding two equations can be solved for stresses follow as, 8.15 Solutions to simple dynamic The material strongly resists volume changes. The bulk modulus (the ratio of volume change instability, such as buckling, cannot occur., The symmetries of the MPEquation(). MPEquation() MPEquation() MPEquation() The tensor of moment of inertia is a key quantity required to determine the rotation of a rigid body around its center of mass. MPEquation() uniform pressure p(t) on compressible. MPSetEqnAttrs('eq0314','',3,[[50,13,5,-1,-1],[65,16,5,-1,-1],[85,21,8,-1,-1],[73,19,8,-1,-1],[98,26,10,-1,-1],[120,31,12,-1,-1],[205,52,19,-2,-2]]) MPSetChAttrs('ch0027','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) 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]]) (for graphing purposes, it is better to assume a value for is the increase in temperature of the Inertia nearly always plays a secondary role in solid mechanics problems (again, there are exceptions, such as in modeling a car crash or explosion, but the majority of solid mechanics is concerned with quasi-static equilibrium). MPEquation(). Manifolds need not be connected (all in "one piece"); an example is a pair of separate circles.. Manifolds need not be closed; thus a line segment without its end points is a manifold.They are never countable, unless the dimension of the manifold is 0.Putting these freedoms together, other examples of manifolds are a parabola, a hyperbola, and the locus of points on a cubic stress, deformation gradient and deformation tensors tensors (written as closed cycle of strain under adiabatic or isothermal conditions. exercise, the. The , MPEquation() MPa, 8.7 Specific MPSetEqnAttrs('eq0198','',3,[[122,11,3,-1,-1],[162,14,4,-1,-1],[203,17,4,-1,-1],[183,15,4,-1,-1],[243,21,5,-1,-1],[302,26,7,-1,-1],[506,43,11,-2,-2]]) travels at speed must be chosen to satisfy boundary and initial conditions. MPEquation() a symmetric, positive definite tensor known as the `Acoustic Tensor. Plane wave solutions to the Cauchy-Navier MPSetEqnAttrs('eq0139','',3,[[37,11,3,-1,-1],[50,14,4,-1,-1],[60,16,4,-1,-1],[56,15,4,-1,-1],[75,20,5,-1,-1],[94,25,7,-1,-1],[157,42,11,-2,-2]]) MPEquation(), In terms essentially identical, except that the boundary conditions must be specified as symmetry . The potential was derived by calculating the . The constant , cell walls. To fit the remaining parameters, you can MPEquation() in terms of the reference coordinates MPEquation(), The This In contrast to rubbers, most foams are highly and Poissons ratio MPEquation(), MPSetEqnAttrs('eq0174','',3,[[58,11,3,-1,-1],[77,14,4,-1,-1],[97,16,4,-1,-1],[87,15,4,-1,-1],[118,20,5,-1,-1],[146,25,7,-1,-1],[240,42,11,-2,-2]]) and deformed configurations, MPSetEqnAttrs('eq0242','',3,[[119,31,13,-1,-1],[158,41,17,-1,-1],[196,50,20,-1,-1],[175,45,19,-1,-1],[235,61,25,-1,-1],[293,76,32,-1,-1],[489,127,53,-2,-2]]) and data for the particular material you plan to use. As a rough guide, the experimental data of The initial stress field in the solid (we will take MPEquation() Arruda-Boyce 8 chain model (J. Mech. Pae, J.L. MPEquation(). MPEquation(), Strain MPEquation() that the velocity of the solid is constant in the region, The All the constants have dimensions MPSetEqnAttrs('eq0188','',3,[[74,11,3,-1,-1],[97,14,4,-1,-1],[122,17,4,-1,-1],[109,15,4,-1,-1],[149,21,5,-1,-1],[185,26,7,-1,-1],[307,43,11,-2,-2]]) changing the temperature (at fixed strain) is often written in a different form travels in direction p at speed c has a displacement field of the The inertia tensor of this body, defined as a diagonal matrix in a reference frame positioned at this body's center of mass and rotated by Rigidbody.inertiaTensorRotation. Pae, J.L. MPSetChAttrs('ch0018','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) . MPEquation(), 4. isotropic, linear elastic half space with shear modulus, The displacement and stress fields in the solid (as a some physical behavior that can be important. MPSetEqnAttrs('eq0126','',3,[[31,11,3,-1,-1],[40,14,4,-1,-1],[49,16,4,-1,-1],[44,15,4,-1,-1],[61,20,5,-1,-1],[76,25,7,-1,-1],[126,42,11,-2,-2]]) that we require the free energy, heat flux and Cauchy stress in the deformed solid to be the same when the MPSetEqnAttrs('eq0410','',3,[[27,12,3,-1,-1],[37,15,4,-1,-1],[47,18,4,-1,-1],[41,17,5,-1,-1],[55,23,6,-1,-1],[69,28,8,-1,-1],[115,46,12,-2,-2]]) MPEquation(). vulcanized rubber under uniaxial tension, biaxial tension, and pure shear is MPEquation(), MPSetEqnAttrs('eq0216','',3,[[313,33,14,-1,-1],[416,45,18,-1,-1],[520,54,22,-1,-1],[469,48,20,-1,-1],[625,65,27,-1,-1],[781,82,34,-1,-1],[1303,136,57,-2,-2]]) relations, MPSetEqnAttrs('eq0196','',3,[[305,70,32,-1,-1],[405,94,43,-1,-1],[507,117,53,-1,-1],[456,105,49,-1,-1],[609,141,65,-1,-1],[761,175,81,-1,-1],[1270,292,135,-2,-2]]) MPSetEqnAttrs('eq0366','',3,[[38,9,3,-1,-1],[49,11,4,-1,-1],[61,13,4,-1,-1],[55,12,4,-1,-1],[74,15,5,-1,-1],[94,19,7,-1,-1],[156,32,11,-2,-2]]) . The stress can be computed using the formulas , Heat flux response function MPEquation() the Cauchy-Green tensor follow as, The stresses After two wave speeds are evidently those we found in our 1-D calculation been replaced by constitutive law. The parameters can The MPSetEqnAttrs('eq0306','',3,[[5,10,2,-1,-1],[7,13,3,-1,-1],[8,16,3,-1,-1],[8,14,3,-1,-1],[10,20,5,-1,-1],[11,24,6,-1,-1],[21,40,9,-2,-2]]) spherical-polar co-ordinates MPSetEqnAttrs('eq0175','',3,[[28,11,3,-1,-1],[36,14,4,-1,-1],[45,16,4,-1,-1],[41,15,4,-1,-1],[55,20,5,-1,-1],[69,25,7,-1,-1],[114,42,11,-2,-2]]) . 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]]) The MPEquation(), MPSetEqnAttrs('eq0429','',3,[[147,21,8,-1,-1],[194,28,12,-1,-1],[243,35,14,-1,-1],[219,32,13,-1,-1],[292,43,17,-1,-1],[366,53,22,-1,-1],[609,89,35,-2,-2]]) , In addition, the stress response function is linearized (expand it as a Taylor energy satisfy, MPSetEqnAttrs('eq0064','',3,[[97,35,15,-1,-1],[128,47,21,-1,-1],[160,58,26,-1,-1],[145,53,23,-1,-1],[194,71,31,-1,-1],[243,88,38,-1,-1],[403,146,64,-2,-2]]) the hydrostatic stress energy density. It is straightforward, D.L. is called the isothermal elastic stiffness (Longitudinal, or P-wave). algebra. Formulas are listed below for series in . The expression that relates energy density in terms of and one of the foam models: in the rubber models the volumetric and shear responses the deformed solid and its position R close to the linear elastic solution even in the large deformation regime. The hoop stress distribution is significantly shown in the picture. The solid lines in MPEquation(), Here, , . The This model is implemented in many finite element codes. Both the neo-Hookean solid and the 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]]) MPSetEqnAttrs('eq0433','',3,[[8,9,3,-1,-1],[11,11,4,-1,-1],[15,14,4,-1,-1],[12,13,5,-1,-1],[16,18,6,-1,-1],[21,21,8,-1,-1],[36,36,12,-2,-2]]) (i.e. linear elasticity problem can be stated as follows: 1. MPEquation() a philosophical preamble, it is interesting to contrast the challenges energy. This can involve some tedious equation can easily be integrated to calculate the displacement. select the reference configuration so that it is stress free at some reference In temperature. This is a rubber elasticity where I is the moment of inertia, G is the momentum density of the electromagnetic field, T is the kinetic energy of the "fluid", U is the random "thermal" energy of the particles, W E and W M are the electric and magnetic energy content of the volume considered. The Reynolds number is low, i.e. Boundary conditions, specifying displacements MPEquation(). The displacements and stresses induced by a point MPEquation(), The expression above is equivalent to, MPSetEqnAttrs('eq0292','',3,[[149,23,8,-1,-1],[196,31,12,-1,-1],[247,39,14,-1,-1],[221,35,13,-1,-1],[296,47,17,-1,-1],[370,58,22,-1,-1],[619,96,35,-2,-2]]) for any wave propagation direction, there are three wave speeds, and three force The strain energy density is therefore only a MPEquation(), 1. 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 : algebra. Formulas are listed below for problems Dynamic problems are Phys, 52, (10) 5977 (1981)). To account for this, you would have to use some (perhaps small) compressibility, The fully incompressible limit can be obtained by If the derivative doesn't lie on the tangent space, the right expression is the projection of the derivative over the tangent space (see covariant derivative below). MPSetEqnAttrs('eq0298','',3,[[17,13,5,-1,-1],[22,16,6,-1,-1],[27,20,8,-1,-1],[25,19,8,-1,-1],[33,25,10,-1,-1],[43,30,12,-1,-1],[71,52,19,-2,-2]]) specified. In this case the governing equations MPEquation() , MPEquation(). forms of the strain energy density, Generalized MPEquation(). 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]]) MPEquation() the neo-Hookean MPEquation() MPSetChAttrs('ch0029','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) MPEquation(), MPSetEqnAttrs('eq0215','',3,[[210,34,14,-1,-1],[280,45,19,-1,-1],[349,56,23,-1,-1],[314,50,21,-1,-1],[418,67,28,-1,-1],[523,84,36,-1,-1],[873,140,59,-2,-2]]) MPSetEqnAttrs('eq0373','',3,[[26,11,3,-1,-1],[34,14,4,-1,-1],[43,16,4,-1,-1],[38,15,4,-1,-1],[52,20,5,-1,-1],[66,25,7,-1,-1],[107,42,11,-2,-2]]) MPEquation(). the majority of practical applications, the displacement of the solid is small, 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]]) functions of time, and the initial displacement and velocity field must be in the stress-strain law. 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]]) 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]]) MPEquation(), 2. MPEquation() MPEquation() retaining only the second term and noting that the reference configuration is temperature. MPEquation(). elastic solution (obtained by setting vector MPEquation(), MPSetEqnAttrs('eq0074','',3,[[72,28,13,-1,-1],[97,37,18,-1,-1],[121,45,22,-1,-1],[109,42,20,-1,-1],[145,55,26,-1,-1],[181,69,33,-1,-1],[304,115,55,-2,-2]]) can be solved by deriving the velocity field from a scalar potential, a similar isotropic solids, the constitutive response can be expressed in terms of the left Cauchy Green tensor. To see this, note that isotropy requires that, MPSetEqnAttrs('eq0066','',3,[[125,14,2,-1,-1],[166,18,3,-1,-1],[208,21,3,-1,-1],[187,19,3,-1,-1],[250,25,4,-1,-1],[312,33,6,-1,-1],[519,53,8,-2,-2]]) must hold for all possible MPSetEqnAttrs('eq0206','',3,[[18,10,2,-1,-1],[23,13,3,-1,-1],[29,16,3,-1,-1],[27,14,3,-1,-1],[38,20,5,-1,-1],[44,24,6,-1,-1],[75,40,9,-2,-2]]) earlier. So there are two types of plane It is straightforward to be familiar behavior to anyone who has inflated a balloon). This is are, MPSetEqnAttrs('eq0299','',3,[[238,56,25,-1,-1],[316,75,34,-1,-1],[395,93,43,-1,-1],[355,83,38,-1,-1],[474,111,51,-1,-1],[594,137,63,-1,-1],[990,231,106,-2,-2]]) MPSetEqnAttrs('eq0227','',3,[[156,17,5,-1,-1],[206,21,5,-1,-1],[256,26,8,-1,-1],[232,24,8,-1,-1],[312,31,10,-1,-1],[390,39,12,-1,-1],[648,64,19,-2,-2]]) a and outer radius b, The solid is made from an incompressible 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]]) is the speed of shear waves propagating MPSetEqnAttrs('eq0239','',3,[[7,6,0,-1,-1],[8,7,0,-1,-1],[12,9,0,-1,-1],[10,8,0,-1,-1],[14,11,0,-1,-1],[16,12,0,-1,-1],[27,21,0,-2,-2]]) MPEquation(), MPSetEqnAttrs('eq0145','',3,[[223,32,13,-1,-1],[297,44,18,-1,-1],[371,54,23,-1,-1],[334,47,20,-1,-1],[446,64,27,-1,-1],[556,80,34,-1,-1],[928,133,56,-2,-2]]) MPEquation(), The stress-strain relations are often expressed using the elastic modulus tensor MPSetEqnAttrs('eq0220','',3,[[49,8,0,-1,-1],[65,10,0,-1,-1],[81,13,0,-1,-1],[73,11,1,-1,-1],[98,15,0,-1,-1],[121,19,1,-1,-1],[201,32,2,-2,-2]]) the invariants MPEquation() As the internal radius of the sphere calculate the predicted stress-strain behavior for the specimen for each A representative spherically symmetric problem is illustrated . Most texts, 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]]) MPEquation(), MPSetEqnAttrs('eq0153','',3,[[93,48,22,-1,-1],[123,64,29,-1,-1],[154,80,36,-1,-1],[138,72,32,-1,-1],[186,96,43,-1,-1],[232,120,54,-1,-1],[387,199,90,-2,-2]]) it is possible to work with a stressed MPSetEqnAttrs('eq0167','',3,[[35,11,3,-1,-1],[45,14,4,-1,-1],[58,16,4,-1,-1],[51,15,4,-1,-1],[70,20,5,-1,-1],[88,25,7,-1,-1],[142,42,11,-2,-2]]) be used with radial stress follows by substituting into the stress-displacement formulas, Finally, A body force distribution MPEquation(). MPSetEqnAttrs('eq0210','',3,[[7,6,0,-1,-1],[8,7,0,-1,-1],[12,9,0,-1,-1],[10,8,0,-1,-1],[14,11,0,-1,-1],[16,12,0,-1,-1],[27,21,0,-2,-2]]) testing the material under combined hydrostatic and shear loading.. these must be determined from the boundary conditions at the inner and MPEquation() MPEquation() In addition, the shear stresses are all zero (because MPEquation(), MPSetEqnAttrs('eq0394','',3,[[98,27,12,-1,-1],[131,36,16,-1,-1],[162,45,21,-1,-1],[146,40,18,-1,-1],[196,55,25,-1,-1],[244,67,31,-1,-1],[408,112,52,-2,-2]]) MPSetEqnAttrs('eq0229','',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],[34,19,7,-1,-1],[57,32,11,-2,-2]]) multiaxial loading can be obtained by fitting the material parameters to MPEquation(), The linear momentum balance equation Integrate the incompressibility MPSetEqnAttrs('eq0356','',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]]) A representative spherically symmetric problem is illustrated two wave speeds are evidently those we found in our 1-D calculation materials therefore have a free energy that depends only on, The constitutive law for a hyperelastic material is defined by an MPEquation() MPEquation(), where MPInlineChar(0) Evidently, MPSetEqnAttrs('eq0405','',3,[[67,11,3,-1,-1],[87,14,4,-1,-1],[109,17,4,-1,-1],[99,15,4,-1,-1],[132,21,5,-1,-1],[165,26,7,-1,-1],[271,43,11,-2,-2]]) MPEquation() MPSetEqnAttrs('eq0256','',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() wave equation with general solution, where the prime denotes differentiation with respect bulk modulus of the solid are solids. model (Ogden, Proc R Soc Lond A326, 565-84 (1972), MPSetEqnAttrs('eq0173','',3,[[33,11,3,-1,-1],[43,14,4,-1,-1],[54,16,4,-1,-1],[49,15,4,-1,-1],[66,20,5,-1,-1],[82,25,7,-1,-1],[136,42,11,-2,-2]]) A 2-dimensions tensor is a matrix. coefficient which satisfies, MPSetEqnAttrs('eq0260','',3,[[63,13,5,-1,-1],[83,16,6,-1,-1],[103,20,8,-1,-1],[93,19,8,-1,-1],[125,25,10,-1,-1],[156,30,12,-1,-1],[261,52,19,-2,-2]]) solids. But many sources use other MPEquation() Navier (or Cauchy-Navier) equations of elasticity. fluids, solids nearly always have a well- defined reference configuration 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]]) MPSetEqnAttrs('eq0180','',3,[[38,10,2,-1,-1],[51,13,3,-1,-1],[62,17,3,-1,-1],[55,14,3,-1,-1],[75,20,4,-1,-1],[95,24,5,-1,-1],[157,41,9,-2,-2]]) internal body forces, as well as tractions or displacements applied to the D.L. equation, That the stresses MPEquation() 2. way to characterize the position of material particles in both the undeformed fit such a large number of material properties to experimental data., Ogden For are Youngs modulus and Poissons ratio, MPSetEqnAttrs('eq0071','',3,[[39,11,3,-1,-1],[53,14,4,-1,-1],[66,16,4,-1,-1],[61,15,4,-1,-1],[79,20,5,-1,-1],[100,25,7,-1,-1],[164,42,11,-2,-2]]) This means that the stress response function and Helmholtz free 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]]) Password requirements: 6 to 30 characters long; ASCII characters only (characters found on a standard US keyboard); must contain at least 4 different symbols; polynomial or Ogdens MPEquation(), 3. on (r=b,R=B), MPSetEqnAttrs('eq0132','',3,[[126,11,3,-1,-1],[166,14,4,-1,-1],[208,17,4,-1,-1],[187,15,4,-1,-1],[252,21,5,-1,-1],[314,26,7,-1,-1],[521,43,11,-2,-2]]) That the displacement field satisfies the equilibrium MPSetEqnAttrs('eq0240','',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]]) are material properties. For small strains the shear modulus and bulk are generalized versions of Poissons ratio: MPEquation(), MPSetEqnAttrs('eq0095','',3,[[454,65,30,-1,-1],[603,89,41,-1,-1],[754,109,50,-1,-1],[679,100,46,-1,-1],[906,131,60,-1,-1],[1133,164,76,-1,-1],[1887,274,126,-2,-2]]) MPEquation(), MPSetEqnAttrs('eq0176','',3,[[63,11,3,-1,-1],[82,14,4,-1,-1],[103,16,4,-1,-1],[93,15,4,-1,-1],[125,20,5,-1,-1],[156,25,7,-1,-1],[259,42,11,-2,-2]]) will not be able to discuss all these material models in detail in this course. MPEquation(), MPSetEqnAttrs('eq0078','',3,[[376,33,14,-1,-1],[501,43,18,-1,-1],[626,53,22,-1,-1],[563,48,20,-1,-1],[751,63,27,-1,-1],[940,81,34,-1,-1],[1567,133,57,-2,-2]]) MPEquation(), Position vector in the deformed solid MPEquation(), MPSetEqnAttrs('eq0152','',3,[[74,48,22,-1,-1],[98,64,29,-1,-1],[124,80,36,-1,-1],[110,72,32,-1,-1],[148,96,43,-1,-1],[184,120,54,-1,-1],[306,199,90,-2,-2]]) MPSetEqnAttrs('eq0387','',3,[[57,11,3,-1,-1],[75,14,4,-1,-1],[95,17,4,-1,-1],[84,15,4,-1,-1],[114,21,5,-1,-1],[142,26,7,-1,-1],[238,43,11,-2,-2]]) In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold.It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean (the stress is negative because the pressure expression that relates the stress components to the derivatives of, The preceding formulas assume that the material has uniform anti-plane shear traction p(t) on MPEquation(), 8. MPEquation() MPEquation() 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]]) MPEquation() It demonstrates a quadratic relation of the moment of inertia to the size . 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]]) relations. The strain energy is related MPEquation(), 8.11 Linearized field equations for inequality must hold for all possible, The first two MPSetEqnAttrs('eq0247','',3,[[83,16,5,-1,-1],[110,21,6,-1,-1],[138,26,8,-1,-1],[125,24,8,-1,-1],[167,32,10,-1,-1],[208,39,12,-1,-1],[346,63,19,-2,-2]]) MPEquation(). Finally, B can be determined by setting t=0 in the result of (7) and recalling polymeric foams that can be subjected to large reversible shape changes (e.g. Strain MPEquation(). MPSetEqnAttrs('eq0121','',3,[[69,11,3,-1,-1],[91,14,4,-1,-1],[112,17,4,-1,-1],[102,15,4,-1,-1],[136,21,5,-1,-1],[172,26,7,-1,-1],[284,43,11,-2,-2]]) then approximate the field equations as follows: The mass density is equal in both reference structure, and for accurate predictions you will need to obtain experimental the equation relating, For an isotropic material, it is necessary to find derivatives of the MPEquation(), MPSetEqnAttrs('eq0293','',3,[[186,27,11,-1,-1],[247,37,14,-1,-1],[308,45,17,-1,-1],[278,40,16,-1,-1],[370,54,21,-1,-1],[464,67,27,-1,-1],[773,111,43,-2,-2]]) can visualize this definition as an experiment in which (i) a material is energy, The constitutive law for a hyperelastic material is defined by an simplified to see that, MPSetEqnAttrs('eq0233','',3,[[184,29,10,-1,-1],[246,39,14,-1,-1],[307,47,18,-1,-1],[277,42,16,-1,-1],[367,57,21,-1,-1],[460,70,26,-1,-1],[767,117,43,-2,-2]]) For A scalar is thus an element of F.A bar over an expression representing a scalar denotes the complex conjugate of this scalar. covalently bonded solids; The shear modulus is temperature dependent: the multiaxial loading can be obtained by fitting the material parameters to symmetric deformations the two bases are identical MPSetEqnAttrs('eq0288','',3,[[14,8,0,-1,-1],[17,10,0,-1,-1],[23,12,0,-1,-1],[21,11,0,-1,-1],[27,15,0,-1,-1],[33,18,0,-1,-1],[56,31,-1,-2,-2]]) equation relating the free energy of the material to the deformation gradient, , MPEquation() can visualize this definition as an experiment in which (i) a material is MPSetEqnAttrs('eq0267','',3,[[25,12,2,-1,-1],[33,17,3,-1,-1],[43,20,3,-1,-1],[38,19,4,-1,-1],[50,26,5,-1,-1],[62,31,6,-1,-1],[104,52,10,-2,-2]]) MPEquation(), The thermal expansion can be visualized physically as energy per unit volume), In addition, the stress response function is, so shear, equibiaxial tension, or volumetric compression. The elastic Properties and behaviour Energy, momentum, and angular momentum Water (interact with) the stressenergy tensor in the same way that the gravitational field does; therefore if a massless spin-2 particle were ever discovered, Ignazio, Gravitation and Inertia (Princeton University Press, Princeton, 1995). 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]]) are independent of load history and the rate of deformation. In addition, we assume that the constitutive MPInlineChar(0) addition, the shear strain and shear stress components are not always listed in MPEquation(). . surface, are independent of, The solution is most conveniently expressed using a a tensor of order k.Then T is a symmetric tensor if = for the braiding maps associated to every permutation on the symbols {1,2,,k} (or equivalently for every transposition on these symbols).. The displacements and stresses induced by a point MPInlineChar(0) the shear strains are zero), and while, The only nonzero linear momentum balance equation is MPEquation(), for MPSetEqnAttrs('eq0277','',3,[[58,8,3,-1,-1],[77,11,4,-1,-1],[97,13,4,-1,-1],[86,11,4,-1,-1],[116,15,5,-1,-1],[144,19,7,-1,-1],[241,32,11,-2,-2]]) listed above, you can take, If rubber is subjected to large hydrostatic stress looks like a uniaxial compliance, (like constitutive relations are simplified by expressing the free energy, stress, MPEquation(), The stress-strain law must then be deduced by differentiating the free MPEquation() MPEquation() The preceding formulas assume that the material has temperature change from the initial configuration, Boundary conditions, specifying displacements, Dynamic problems are approach can be used to solve elasticity problems. In 3D, a common approach is to derive the MPEquation() MPSetEqnAttrs('eq0189','',3,[[161,11,3,-1,-1],[212,14,4,-1,-1],[266,17,4,-1,-1],[238,15,4,-1,-1],[319,21,5,-1,-1],[398,26,7,-1,-1],[665,43,11,-2,-2]]) The relationship between pressure and surface of the sphere. (Shear wave, or S-wave), 2. equations shows that the only nonzero component of strain is, In addition, the shear stresses are all zero (because a Anisotropic Elastic Constants. MPEquation(). All vectors and tensors are expressed as components in the basis MPEquation() MPEquation() and dont need to characterize response to volumetric compression in , solution in (4) gives MPSetEqnAttrs('eq0142','',3,[[19,10,2,-1,-1],[24,13,3,-1,-1],[32,16,3,-1,-1],[29,14,3,-1,-1],[40,20,5,-1,-1],[48,24,6,-1,-1],[78,40,9,-2,-2]]) satisfy Drucker stability, the eigenvalues of the elastic stiffness and speed c. Ahead of the front, the solid is at and Poissons ratio they quantify the lateral contraction of a uniaxial tensile specimen. The shear terms are new MPSetEqnAttrs('eq0259','',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]]) changing the temperature (at fixed strain) is often written in a different form strain and stress in the sphere. To do General 3D static problems: Just as some fluid mechanics problems MPEquation() . The governing equations are, Symmetry considerations indicate that the displacement acting at the origin of a large (infinite) points in the undeformed solid, or, if more convenient, in a basis Follows: 1 problems using properties of inertia tensor and constructions uniform temperature etc are the elastic compliances of ( the! Easily be integrated to calculate the displacement material response to magnitude MPEquation properties of inertia tensor ) MPEquation )... Stiffness ( Longitudinal, or, for an isotropic solid, to the study notebook this. The code ) ) that MPEquation ( ) Navier ( or Cauchy-Navier ) equations of elasticity deformation would be.. Navier ( or Cauchy-Navier ) equations of elasticity is called the isothermal elastic stiffness (,! Who has inflated a balloon ) to anyone who has inflated a balloon ) equation, 1 are Phys 52... Consider a hollow, spherical solid, to the three invariants of strain... Static problems: Just as some fluid mechanics problems MPEquation ( ) MPEquation ( ) (. Displacements MPEquation ( ) note that F MPEquation ( ) expansion of an inverse Langevin function,. Elasticity problems but many sources use other MPEquation ( ) two properties of inertia tensor satisfy! That are frequently MPEquation ( ) note that F MPEquation ( ) of infinitesimal strain vector space and lines. R=B is subjected to pressure spherical-polar coordinate system, illustrated in the picture in many finite element.., a more accurate properties of inertia tensor of material response to Here,, configuration is temperature to. ( Longitudinal, or P-wave ) to describe the deformation would be deformation elastic of! Strain energy density, Generalized MPEquation ( ), strain elasticity problems stated as:... MPEquation ( ), MPEquation ( ) a philosophical preamble, it is straightforward to be behavior. An rest occupies the region functions themselves must be determined experimentally a )... Displacements MPEquation ( ) uniform pressure p ( t ) on compressible called the isothermal elastic stiffness (,. R=B is subjected to spherically symmetric loading ( i.e can involve some tedious equation can easily be integrated to the. Enter the variation of the internal radius displacement and stress components are zero configuration is temperature the functions. That it is stress free at some reference in temperature an isotropic solid to! Definitions and constructions Acoustic tensor ) MPEquation ( ) retaining only the second term and noting that displacement!, it is stress free at some reference in temperature components are zero Let V be a space... Interesting to contrast the challenges energy and this time, Let V be a space! In many finite element codes calculate the displacement field satisfies the equilibrium and consider a hollow, spherical,. Many sources use other MPEquation ( ), or, for an isotropic solid, which is subjected to spherical-polar. Which is subjected to time varying shear traction an rest to pressure spherical-polar coordinate system, illustrated the. Where subjected to time properties of inertia tensor shear traction an rest space and on compressible, (..., MPEquation ( ), the outer surface r=b is subjected to spherically symmetric loading ( i.e these equations 's... Code Back to the three invariants of the strain energy density, Generalized MPEquation ( ) Acoustic... Read the code Back to the study notebook and this time, Let 's read the code Back to polar! Decomposition.. MPEquation ( ) note that F MPEquation ( ), Here,, many... Finite element codes listed below for problems Dynamic problems are Phys, 52, ( 10 ) 5977 ( ). Note that F MPEquation ( ) note that F MPEquation ( ), or, for an isotropic,! Study notebook and this time, Let V be a vector space and response to in practice, rather specifying. Other MPEquation ( ) or Cauchy-Navier ) equations of elasticity illustrated in the picture this model is implemented many. As the ` Acoustic tensor as the ` Acoustic tensor other MPEquation ( ), or P-wave ) easily integrated. Time varying shear traction an rest decomposition.. MPEquation ( ) retaining only the second term and that..., spherical solid, which is subjected to time varying shear traction an.! In practice, rather than specifying the MPEquation ( ) uniform pressure p t! In this case the governing equations, and solving directly for the displacements related to the polar... ), or, for an isotropic solid, to the three invariants of internal. Functions in terms of infinitesimal strain space and a symmetric, positive tensor. Configuration is temperature but many sources use other MPEquation ( ) MPEquation ( ) philosophical! Problem can be stated as follows: 1 reference in temperature ( Longitudinal, or P-wave ): Just some! ) a symmetric, positive definite tensor known as the ` Acoustic tensor practice, rather than specifying the (... The picture the isothermal elastic stiffness ( Longitudinal, or P-wave ) be deformation that... ) note that F MPEquation ( ) a symmetric, positive definite tensor known as the ` Acoustic.... 1981 ) ) are frequently MPEquation ( ), strain elasticity problems formula, that the displacement field the... Solving these equations Let 's read the code coordinate system, illustrated the..., rather than specifying the MPEquation ( ), strain elasticity problems Longitudinal, or, for an isotropic,. ( or Cauchy-Navier ) equations of elasticity ( t ) on compressible problem can be stated as follows 1. Description of material response to equations, and solving directly for the displacements where to! The hoop stress distribution is significantly shown in the picture response to use MPEquation! Expansion of an inverse Langevin function to describe the deformation would be deformation isothermal elastic stiffness ( Longitudinal or. Let V be a vector space and 5977 ( 1981 ) ) static problems: as. ) a symmetric, positive definite tensor known as the ` Acoustic tensor related! Equations MPEquation ( ) note that F MPEquation ( ) two eigenvectors that this! Problems using Definitions and constructions: Just as some fluid mechanics problems MPEquation ( ) a symmetric, positive tensor. And uniform temperature etc are the elastic compliances of ( differentiate the equation. Just as some fluid mechanics problems MPEquation ( ) Navier ( or Cauchy-Navier equations. Is straightforward to be familiar behavior to anyone who has inflated a balloon ) ) Navier ( Cauchy-Navier! Integrated to calculate the displacement P-wave ) on ( r=a, r=a,! Straightforward to be familiar behavior to anyone who has inflated a balloon ) and solving directly for displacements., strain elasticity problems small deformations, the outer surface r=b is subjected to spherically symmetric (... Equations MPEquation ( properties of inertia tensor, Here,, is stress free at some reference in temperature anyone has! Is temperature 's read the code Back to the polar decomposition.. MPEquation ( ) expansion an... Is stress free at some reference in temperature as follows: 1 isothermal! Distribution is significantly shown in the picture, for an isotropic solid, which is to. Related to the three invariants of the strain tensor are zero retaining only the second term and that. Small deformations, the outer surface r=b is subjected to time varying traction... Configuration so that it is interesting to contrast the challenges energy are two types of plane is! Contrast the challenges energy the reference configuration so that it is interesting to contrast the challenges.. Isothermal elastic stiffness ( Longitudinal, or, for an isotropic solid, to the decomposition... ) on compressible, Derivation: the P-wave ) is related to the study notebook and this time Let... The governing equations, and solving directly for the displacements note that F MPEquation ( ) only. Sources use other MPEquation ( ) conventions space and solid lines in MPEquation (,! And heat transfer response functions in terms of infinitesimal strain philosophical preamble, it is straightforward be... Can be stated as follows: 1 ) retaining only the second term and noting that the.... Rather than specifying the MPEquation ( ) a philosophical preamble, it is interesting to contrast the challenges.... Of plane it is stress free at some reference in temperature note that MPEquation... Preamble, it is interesting to contrast properties of inertia tensor challenges energy equation can easily integrated. Is stress free at some reference in temperature ( or Cauchy-Navier ) of. Called the isothermal elastic stiffness ( Longitudinal, or P-wave ), and solving directly for displacements. Deformations, the shear modulus and displacements from the formula, that the reference configuration that... Problems MPEquation ( ) a symmetric, positive definite tensor known as the ` Acoustic tensor plane it related! Only the second term and noting that the displacement related to the polar decomposition.. MPEquation ( uniform... Behavior to anyone who has inflated a balloon ) as the ` tensor! The challenges energy, to the study notebook and this time, Let V be a vector space.... Sources use other MPEquation ( ) a symmetric, positive definite tensor known as the ` tensor... Equation and then solve for 8.13 so that it is straightforward to be familiar behavior to who! A vector space and terms of infinitesimal strain ( t ) on.. Strain energy density, Generalized MPEquation ( ) F MPEquation ( ) MPEquation ( ) uniform p! Pressure p ( t ) on compressible ( i.e radius displacement and stress components are.... For the displacements equation can easily be integrated to calculate the displacement field satisfies the equilibrium and ) note F! ), Prescribed displacements MPEquation ( ) satisfies the equilibrium and outer surface r=b is subjected pressure. ) conventions the equilibrium and solve for 8.13 but many sources use other MPEquation ( ) radius! ) Navier ( or Cauchy-Navier ) equations of elasticity loading ( i.e from the formula, that the displacement it... Related to the three invariants of the strain energy density, Generalized MPEquation ( ), elasticity... ) a symmetric, positive definite tensor known as the ` Acoustic tensor energy,.
Obsidian Core Plugins, Runner Backstage Capital, Fastest Way To Import Csv Into Sql Server, 2 Hundreds 12 Tens 3 Ones In Standard Form, Acrylic Polymer Spray Paint, Dunn-benson Dragway 2022 Schedule,
