StressStrainTensor

E106 Stress and Strain Tensor Summary Page 1 Stress and Strain Tensors Stress at a point. Imagine an arbitrary solid body oriented in a cartesian coordinate system. A number of forces are acting on this body in different directions but the net force (the vector sum of the forces) on the body is 0. Conceptually slice the body on a plane normal to the x -direction (parallel to the yz -plane). Take a small area on this plane and call it . Calculate the resolved force acting on this small area and call it . . Notice that since is the total force acting only on , the magnitude of will change as changes. We can define three scalar quantities, , , and . The first subscript refers to the plane and the second refers to the force direction. If we do the same conceptual experiment at the same location but in the y and z -direc- tions, we obtain , , , , , and . For static equilibrium , , and , resulting in six inde- pendent scalar quantities. These six scalars arranged in an ordered matrix forms the stress tensor , .A x/H9004 Axxˆ /H9004/H11005 F/H9004 F/H9004 F/H9004x F/H9004y F/H9004z /H11001/H11001 F/H9004xxˆ F/H9004yyˆ F/H9004zzˆ /H11001/H11001 /H11005/H11005 F/H9004 Ax/H9004 F/H9004 Ax/H9004 /H9268xxF/H9004x Ax/H9004----------- Ax/H9004 0→lim /H11005 /H9268xy /H9270xyF/H9004y Ax/H9004----------- Ax/H9004 0→lim /H11005/H11005 /H9268xz /H9270xzF/H9004z Ax/H9004----------- Ax/H9004 0→lim /H11005/H11005 /H9268yyF/H9004y Ay/H9004----------- Ay/H9004 0→lim /H11005 /H9268yx /H9270yxF/H9004x Ay/H9004----------- Ay/H9004 0→lim /H11005/H11005 /H9268yz /H9270yzF/H9004z Ay/H9004----------- Ay/H9004 0→lim /H11005/H11005 /H9268zzF/H9004z Az/H9004----------- Az/H9004 0→lim /H11005 /H9268zx /H9270zxF/H9004x Az/H9004----------- Az/H9004 0→lim /H11005/H11005 /H9268zy /H9270zyF/H9004y Az/H9004----------- Az/H9004 0→lim /H11005/H11005 /H9268xy /H9268yx /H11005 /H9268yz /H9268zy /H11005 /H9268xz /H9268zx /H11005 33/H11003 /H9268 /H9268ij/H9268xx/H9268xy/H9268xz /H9268xy/H9268yy/H9268yz /H9268xz/H9268yz/H9268zz/H11005/H11005 E106 Stress and Strain Tensor Summary Page 2For example, the stress tensor for a cylinder with cross-sectional area in uniaxial tension from force is if the cylinder axis and are both parallel to the x-axis, if the cylinder axis and are both parallel to the y-axis, and if the cylinder axis and are both parallel to the z-axis. The sign convention for the stress elements is that a positive force on a positive face or a negative force on a negative face is positive. All others are negative. As a final example, a cube oriented so that its faces are perpendicular to the coordinate axes, with an area per face of has the following forces applied to it: force applied to the positive x face in the positive x direction, force applied to the positive y face in A0 F /H9268 /H9268ijF A0-------00 00 0 00 0/H11005/H11005 F /H9268 /H9268ij000 0F A0-------0 000/H11005/H11005 F /H9268 /H9268ij00 0 00 0 00F A0-------/H11005/H11005 F A0 F1 F2 E106 Stress and Strain Tensor Summary Page 3the positive y direction, and force applied to the positive x face in the negative y direction. The necessary forces to keep the cube form moving are applied to the other faces. The resultant stress tensor is . Stress on a plane. It is often necessary to calculate the stress on an arbitrarily-oriented plane with nor- mal inside a solid. A force balance on the tetrahedron formed by the intersection of the plane with the coordinate axes provides the needed results. We define a stress vector, , defined as limit of the net force acting on the plane, , per unit area as the area shrinks to zero. This vector can be decomposed into the normal stress on the plane (the force per unit area in the direction normal to the plane), , and the shear stress on the plane (the force per unit area in a direction lying in the plane), . It can also be decomposed into components in the three coordinate-axis directions, , , and . In other words, . If we define the direction cosines of as , , and , then , , and ,F3 /H9268 /H9268ijF1 A0-------F3/H11002 A0------------0 F3/H11002 A0------------F2 A0-------0 00 0/H11005/H11005 nˆ sF sn ss sx sy sz ssxsysz /H11001/H11001 sxxˆsyyˆszzˆ /H11001/H11001 /H11005/H11005 nˆ k nˆxˆ• /H9251cos /H11005/H11005 l nˆyˆ• /H9252cos /H11005/H11005 m nˆzˆ• /H9253cos /H11005/H11005 sx /H9268xxk /H9268xyl/H9268xzm /H11001/H11001 /H11005 sy /H9268xyk /H9268yyl/H9268yzm /H11001/H11001 /H11005 sz/H9268xzk /H9268yzl/H9268zzm /H11001/H11001 /H11005 E106 Stress and Strain Tensor Summary Page 4or . We can then find the normal component of the stress, , by . The shear component can be determined but requires a little more work. sx sy sz/H9268ijnˆ/H9268xx/H9268xy/H9268xz /H9268xy/H9268yy/H9268yz /H9268xz/H9268yz/H9268zzk l m/H11005/H11005 sn snsnˆ• sxsyszk l m/H11005/H11005 E106 Stress and Strain Tensor Summary Page 5Coordinate transformations and stress invariants. It is often useful to know the stress tensor in a coordinate system that has been rotated and/or translated with respect to the original coordinate system. We can transform the coordinates, into the coordinates, by use of a transformation matrix, , where has the property . In particular, a rotation about the -axis through an angle is given by . To get all of the elements of the stress tensor in the new coordinate system, . The above relationship is often used to define a tensor of rank 2. Several properties of the stress tensor remain unchanged by a change in coordinates. These properties are called invariants . These invariants are closely related to important quantities. The first invariant , , is the trace of the matrix, . The hydrostatic component of (the part due to uniform pressure on all exterior surfaces of the solid) is equal to .The second invariant , is given by . The third invariant is . A common question in stress problem is: Is there a coordinate system for which all of the shear stresses disappear, and the remaining stresses are purely tensile or com-xyz Tx/H11032y/H11032z/H11032T QQ Q1/H11002QT/H11005 z /H9258 x/H11032 y/H11032 z/H11032/H9258cos /H9258 sin 0 /H9258 sin /H11002 /H9258 cos 0 00 1 x y z /H11080 /H11005 /H9268/H11541 Q/H9268QT/H11005 I1 I1 /H9268xx /H9268yy /H9268zz /H11001/H11001 /H11005 /H9268ij I13/H11408 I2 I2 /H9268xy2/H9268yz2/H9268zx2/H9268xx/H9268yy /H11002 /H9268yy/H9268zz /H11002 /H9268zz/H9268xx /H11002 /H11001/H11001 /H11005 I3 /H9268xx/H9268yy/H9268zz2/H9268xy/H9268yz/H9268zx /H9268xx/H9268yz2 /H11002 /H9268yy/H9268zx2 /H11002 /H9268zz/H9268xy2 /H11002 /H11001 /H11005 E106 Stress and Strain Tensor Summary Page 6 pressive? It turns out that there is. The resultant stresses are called the principal stresses , the planes on which they occur are the principal planes , and the directions of the resultant force components are the principal directions or principal axes . If we call the principal stresses , , and , then the problem appears as: Are there values of for which ? The principal stresses are the eigenvalues and the principal directions are the eigen- vectors. The eigenvalue problem can be rewritten in terms of the three invariants as . For any stress tensor, three real (but possibly not distinct) roots will result. The Von Mises yielding criterion. In a complex stress field it is not easy to determine if the stress has exceeded the yield stress in the body. Von Mises proposed the following criterion: Yielding occurs when the second invariant of the stress deviator exceeds some critical value.The stress deviator is the stress tensor with the hydrostatic component removed, i.e. , where . The second invariant of is , which must be greater than some constant for yielding to occur. In terms of the yield stress, , the criterion is/H92611/H92612 /H92613 /H9261 Det/H9268xx/H9261/H11002 /H9268xy/H9268xz /H9268xy/H9268yy/H9261/H11002 /H9268yz /H9268xz /H9268yz /H9268zz /H9261/H110020/H11005 /H92613I1/H92612 /H11002 I2/H9261 /H11002 I3/H11002 0/H11005 /H9268ijD/H9268ijhI/H11002/H9268xx h/H11002 /H9268xy /H9268xz /H9268xy /H9268yy h/H11002 /H9268yz /H9268xz /H9268yz /H9268zzh/H11002/H11005/H11005 h/H9268xx /H9268yy /H9268zz /H11001/H11001 3----------------------------------------- - /H11005 /H9268ijD I2/H9268xx /H9268yy/H11002 ()2 /H9268yy /H9268zz/H11002 ()2 /H9268zz /H9268xx/H11002 ()26/H9268xy2/H9268yz2/H9268xz2/H11001/H11001 () /H11001/H11001/H11001 6------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------ /H11005 k2 /H9268y E106 Stress and Strain Tensor Summary Page 7for yielding to occur. The Strain Tensor.Normal strain is the change in length in a given direction divided by the initial length in that direction. Shear strain is the complement of the angle between two initially perpendicular line segments. If you apply a force to a solid object you may end up simultaneously translating, rotating and deforming the object. The vector function which describes the difference between the initial position and the final position of each point in the object is . If we take the gradient of u we end up with a tensor . For small strains we find that where . The diagonal terms are the normal strains in the x, y, and z directions respectively. The off-diagonal terms are equal to one-half of the engineering shear strain, e.g., . In terms of ,/H9268xx /H9268yy/H11002 ()2/H9268yy /H9268zz/H11002 ()2/H9268zz /H9268xx/H11002 ()26/H9268xy2/H9268yz2/H9268xz2/H11001/H11001 () /H11001/H11001/H11001 2------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------12/ /H9268y/H11350 uxyz,,()ux y z,,() vx y z,,() wx y z,,()/H11005 /H11612ux/H11128/H11128u y/H11128/H11128u z/H11128/H11128u x/H11128/H11128v y/H11128/H11128v z/H11128/H11128v x/H11128/H11128w y/H11128/H11128w z/H11128/H11128w/H11005 /H11612u /H11612uT/H11001 2/H9280 /H11005 /H9280 /H9280kl/H9280xx/H9280xy/H9280xz /H9280yx/H9280yy/H9280yz /H9280zx/H9280zy/H9280zz/H11005/H11005 /H9280xy/H9253xy2/H11408 /H11005 u/H11612 E106 Stress and Strain Tensor Summary Page 8, , and . As was the case with stress , , and . We can determine the strains in a rotated coordinate system in the same way as for stresses. We can transform the coordinates, into the coordinates, by use of a transformation matrix, , where has the property . To get all of the elements of the strain tensor in the new coordinate system, . Relationship between stress and strain. Every member of will cause a corresponding stress in . The relationship can be written as . Writing out the first term explicitly should suffice to explain the notation. . Fortunately only 21 of the 81 -terms are unique. To simplify the notation, the stress and strain tensors are rewritten as vectors. The simplified notation is known as contracted notation . First the off-diagonal strain terms are converted to engineer- ing shear strains. . The resulting matrix is no longer a tensor because it doesn’t follow the coordinate- transformation rules. Then the elements are renumbered./H9280xy1 2--- y/H11128/H11128u x/H11128/H11128v/H11001 /H11005 /H9280xz1 2--- z/H11128/H11128u x/H11128/H11128w/H11001 /H11005 /H9280yz1 2--- z/H11128/H11128v y/H11128/H11128w/H11001 /H11005 /H9280yx /H9280xy/H11005 /H9280yz /H9280zy/H11005 /H9280xz /H9280zx/H11005 xyzTx/H11032y/H11032z/H11032T QQ Q1/H11002QT/H11005 /H9280/H11541[] Q/H9280[]QT/H11005 /H9280kl /H9268ij /H9268ij Cijkl/H9280kl /H11005 /H9268xxCxxxx/H9280xxCxxxy/H9280xyCxxxz/H9280xz/H11001/H11001 /H11005 Cxxyx/H9280yx Cxxyy/H9280yy Cxxyz/H9280yz /H11001/H11001/H11001 Cxxzx/H9280zxCxxzy/H9280zyCxxzz/H9280zz/H11001/H11001/H11001 Cijkl /H9280xx2/H9280xy2/H9280xz 2/H9280yx/H9280yy2/H9280yz 2/H9280zx2/H9280zy/H9280zz/H9280xx/H9253xy/H9253xz /H9253yx/H9280yy/H9253yz /H9253zx/H9253zy/H9280zz/H11005 E106 Stress and Strain Tensor Summary Page 9, . Then the matrices are written as vectors, , . Finally the relationships between the stress vector and the strain vector is expressed. . The materials-property matrix with all of the Q’s is known as the stiffness matrix . Unfortunately is used for both the stiffness matrix and the coordinate transfor- mation matrix. Don’t get them confused. The stiffness matrix is used when all of the strains are known and the values of the stresses are to be determined. In the more common case of the stresses being known and the strains to be determined, the inverse of the stiffness matrix, called the compliance matrix , , must be used. The relationship between and is that . There are a number of simpli- fied cases for the stiffness and compliance matrices./H9268xx/H9268xy/H9268xz /H9268xy/H9268yy/H9268yz /H9268xz/H9268yz/H9268zz/H92681/H92686/H92685 /H92686/H92682/H92684 /H92685/H92684/H92683/H11005/H9280xx/H9253xy/H9253xz /H9253yx/H9280yy/H9253yz /H9253zx/H9253zy/H9280zz/H92801/H92806/H92805 /H92806/H92802/H92804 /H92805/H92804/H92803/H11005 /H92681/H92686/H92685 /H92686/H92682/H92684 /H92685/H92684/H92683/H92681 /H92682 /H92683 /H92684 /H92685 /H92686⇒/H9268xx /H9268yy /H9268zz /H9268yz /H9268xz /H9268xy/H11005/H92801/H92806/H92805 /H92806/H92802/H92804 /H92805/H92804/H92803/H92801 /H92802 /H92803 /H92804 /H92805 /H92806⇒/H9280xx /H9280yy /H9280zz /H9253yz /H9253xz /H9253xy/H11005 /H92681 /H92682 /H92683 /H92684 /H92685 /H92686Q11Q12Q13Q14Q15Q16 Q12Q22Q23Q24Q25Q26 Q13Q23Q33Q34Q35Q36 Q14Q24Q34Q44Q45Q46 Q15Q25Q35Q45Q55Q56 Q16Q26Q36Q46Q56Q66/H92801 /H92802 /H92803 /H92804 /H92805 /H92806/H11005 Q S QS S Q1/H11002 /H11005 E106 Stress and Strain Tensor Summary Page 10. Linear elastic isotropic materials : The simplest materials are ones in which the properties do not vary with direction, or linear elastic isotropic materials. In a linear elastic isotropic material character- ized by Young’s modulus, , Poisson’s ratio, , and the shear modulus, , the following relationships between the stress tensor and the strain tensor hold: , , and , also , , and/H92801 /H92802 /H92803 /H92804 /H92805 /H92806S11S12S13S14S15S16 S12S22S23S24S25S26 S13S23S33S34S35S36 S14S24S34S44S45S46 S15S25S35S45S55S56 S16S26S36S46S56S66/H92681 /H92682 /H92683 /H92684 /H92685 /H92686/H11005 E /H9263 G E 21 /H9263/H11001()----------------------/H11005 /H9280xx1 E----/H9268xx /H9263/H9268yy /H9268zz/H11001 ()/H11002 [] /H11005 /H9280yy1 E----/H9268yy /H9263/H9268xx /H9268zz/H11001 ()/H11002 [] /H11005 /H9280zz1 E----/H9268zz /H9263/H9268xx /H9268yy/H11001 ()/H11002 [] /H11005 /H9253xy 2/H9280xy/H9268xy G--------/H11005/H11005 /H9253xz 2/H9280xz/H9268xz G--------/H11005/H11005 /H9253yz 2/H9280yz/H9268yz G--------/H11005/H11005 E106 Stress and Strain Tensor Summary Page 11It is left to the reader (usually on an exam) to wade through the alphabet soup and determine the stiffness and compliance matrices for a linear elastic isotropic mate-rial.Linear elastic orthotropic materials : For an orthotropic material (one in which the properties in the y- and z-directions are the same but different in the x-direction, such as a composite material with the fibers all oriented in the x-direction) the stiffness matrix has the form , and the compliance matrix has the form . Since the properties in the y and z directions are equal, the two-dimensional x,y-case is often considered when determining materials properties. In this case the stiffness-matrix relationship becomes and the compliance-matrix relationship becomesQQ 11Q12Q13000 Q12Q22Q23000 Q13Q23Q33000 000 Q4400 0000 Q550 00000 Q66/H11005 SS11S12S13000 S12S22S23000 S13S23S33000 000 S4400 0000 S550 00000 S66/H11005 /H9268xx /H9268yy /H9268xy Q11Q120 Q12Q220 00 Q66/H9280xx /H9280yy /H9253xy /H11005 E106 Stress and Strain Tensor Summary Page 12. First a uniaxial tensile test is performed in the x-direction. is applied and , and are measured. As always in a uniaxial tensile test, , and , where is the elastic modulus in the longitudinal or x direction, and is Poisson’s ratio for stress in the longitudinal direction and strain in the trans- verse direction. By comparison with the compliance matrix, it is seen that , and . Next a uniaxial tensile test is performed in the y-direction. is applied and , and are measured. As before, , and , where is the elastic modulus in the transverse or y direction, and is Poisson’s ratio for stress in the transverse direction and strain in the longitudinal direction. By comparison with the compliance matrix, it is seen that , and . A final test with pure shear gives the relationship . The interested reader should be able to determine the relationship between and /H9280xx /H9280yy /H9253xy S11S120 S12S220 00 S66/H9268xx /H9268yy /H9268xy /H11005 /H9268xx /H9280xx /H9280yy /H9268xx Ecl/H9280xx /H11005 /H9263lt/H9280yy /H9280xx-------- /H11002/H11005 Ecl /H9263lt S111 Ecl--------/H11005 S12/H9263lt Ecl--------/H11002/H11005 /H9268yy /H9280xx /H9280yy /H9268yy Ect/H9280yy /H11005 /H9263tl/H9280xx /H9280yy-------- /H11002/H11005 Ect /H9263tl S221 Ect--------/H11005 S12/H9263tl Ect--------/H11002/H11005 S661 G----/H11005 /H9263lt E106 Stress and Strain Tensor Summary Page 13 from the relationships and . The structure of is not nearly so simply related to , , , and as is , but, as before, it can be determined from , or ./H9263tl S12/H9263lt Ecl--------/H11002/H11005 S12/H9263tl Ect--------/H11002/H11005 Q EclEct/H9263lt /H9263tl S SQ I /H11005 SQ1/H11002/H11005