SQEP Tutorial Session 5: T7S0 Relates to Knowledge & Skills.5,.8 Last Update: //3 Force on an element of area; Definition of principal stresses and strains; Definition of Tresca and Mises equivalent stresses; Mohr s circles; Hdrostatic and deviatoric stresses; How is an element of area described? An element of area is described b a vector. The magnitude of the vector equals the area. The direction of the vector is the normal to area element. Note that this means that area elements have an orientation, i.e., an outside and an inside. A Given the stress matri, what is the force acting on an element of area? If the area happened to be oriented in the -direction, the three components of the force would be, b definition, A, A and A. For an other orientation of the area, if A is the component of the area-vector in the -direction, then this component of the area produces force components of A, A and A. Similarl, the component of the area-vector in the -direction, A, produces force components A, A and A. The total force in the -direction is thus, z F = A + A + A z z z A And similarl in the other directions, so the overall result is epressed compactl as, F = j A i (Note the summation convention for repeated indices). If we adopt vector/matri notation this can also be written, i i F= ( )A (because is smmetric) () Qu. What is the direct stress in an arbitrar direction, nˆ? For a unit area, we can put A= nˆ, where nˆ is the unit vector normal to the area element. The force acting on this unit area is thus F= ( )nˆ, so the component of this T force in the direction of nˆ is F = nˆ ( )nˆ. But this is the normal force on a unit area, n i.e., it is the direct stress in the direction nˆ. That is nn= ˆ ( )nˆ, or, in component notation, nn i j n T = nˆ nˆ () z
How do all the stress components change due to a rotation of the coordinate sstem? Restricting attention to a rotation in the (,) plane, the stress components in the rotated coordinate sstem are given b, τ τ cosθ = sinθ sinθ cosθ τ τ cosθ sinθ sinθ cosθ (this will be derived in Session 6) and hence, = cos θ + sin θ + τ cosθsin θ = cos θ + sin θ τ cosθsin θ (4) τ = τ cos θ + cosθsin ( ) θ What is meant b the principal stresses? Supposing the element of area is rotated until the force acting on it is normal to the surface, i.e., F is parallel to A (i.e., normal to the element of area). Then the stress in this direction is one of the principal stresses. Three. How man principal stresses are there? The directions in which the principal stresses act are mutuall orthogonal, i.e. the form a Cartesian coordinate sstem. So, it is alwas possible to rotate the initial (,,z) coordinate sstem to align with the principal aes. The proof of these assertions will be given in Session 6. What does the stress matri look like in principal coordinates? Suppose (, z ), are the principal aes. The forces acting on the faces of a unit cube oriented parallel to these aes are all perpendicular to the faces. In other words, there are onl direct stresses, all the shear stresses being zero. So the stress matri is diagonal, ( ) (3) 0 0 = 0 0 (5) 0 0 3 NB: It is conventional to order the principal stresses so that > > 3, but their order in the matri ma not be as indicated above. Can the principal aes be defined b the vanishing of the shear stresses? Yes. The principal aes are the onl aes for which the shears all disappear. To be pedantic, if two of the principal stresses are equal then there is a degree arbitrariness in the orientation of the principal aes (degenerac). Are there principal strains also? Yes. For an isotropic, elastic material this follows from the 3D Hooke s law, since the vanishing of the shear stresses implies the vanishing of the shear strains, i.e., because G γ =, etc. Moreover, in this case the principal stress aes and the principal strain aes are the same.
Yes. Do principal aes alwas eist even for anisotropic materials? The eistence of principal stress directions (in terms of which all shear stresses are zero) follows as a mathematical consequence of the smmetr of the stress tensor. Eactl the same is true for strain. The eistence of principal strain directions (in terms of which all shear strains are zero) follows as a mathematical consequence of the smmetr of the strain tensor. Consequentl, principal aes for both stress and strain alwas eist, irrespective of isotrop or elasticit. No. Are the principal aes alwas the same for stress and strain? In linear elasticit for an isotropic material the principal aes for stress coincide with those for strain, since the vanishing of shear stresses implies the vanishing of shear strains in this case. However, for anisotropic materials the principal stress aes and the principal strain aes will generall be different. What are equivalent stresses? Equivalent stresses are a convenient wa of reducing the si components of the stress matri to a single scalar measure of (roughl speaking) the severit of the stressing. More accuratel, the epress the severit of the stressing onl in certain respects, such as the proimit to, or degree of, ielding, or the degree of distortion. Be aware, though, that other stress measures, independent of the equivalent stress, such as the hdrostatic stress, can strongl influence other issues (such as fracture). What is the Tresca stress? The Tresca stress is simpl twice the maimum shear stress for an orientation of the coordinate sstem. It is given in terms of the maimum and minimum principal stresses b, Tresca = 3 = twice maimum shear (6) Wh is the maimum shear equal to ( )/? 3 Recall from Session 4 that a pure shear of τ corresponds to in-plane principal stresses of τ and τ. If we have in-plane principal stresses and, then the D (in-plane) stress matri can be epressed as the sum, 0 0 = + 0 0 + + 0 0 ( ) But the first term on the RHS is an isotropic stress, and hence does not give rise to an shear when rotated (it is a purel hdrostatic stress). The second term, however, is of pure shear form or rather it becomes so when rotated b 45 o and the magnitude of the shear is just ( )/. Of the three coordinate planes, the largest shear is therefore found between the /. QED. maimum and the minimum principal stresses, i.e., ( ) 3 (7)
What is the von Mises stress? In terms of the principal stresses the Mises stress is given b, [( ) + ( ) + ( ) ] = 3 3 (8) In an arbitrar coordinate sstem, the Mises stress is, [( ) + ( z ) + ( z ) + 6( + z z )] = + (9) The main motivation is that setting the Mises stress to the ield stress provides the best ield criterion for multi-grained, isotropic metals (well, structural steels, at least). Do the Mises or Tresca stresses depend upon the coordinate sstem used? No. In other words, the Mises stress and the Tresca stress are scalars. This is proved in Session 8b. What is the hdrostatic stress? The hdrostatic stress is the average of the three direct stresses, i.e., one-third of the trace of the stress matri. Note that it is also a scalar, i.e., it is the same in all coordinate sstems. ( ) 3 = / 3 = ( + + )/ 3 = ( + )/ 3 H = Tr / ii 3 + zz (0) If a stress state is changed b adding a purel hdrostatic stress, do the equivalent stresses change? No. Neither the Tresca stress nor the Mises stress change. This is because all three principal stresses increase b the same amount, i.e., b the hdrostatic stress added. What are the deviatoric stresses? The deviatoric stresses are the on-diagonal (direct) stresses minus the hdrostatic stress. The are denoted b a hat (caret): thus, ˆ = H. The shear stresses are not affected. Hence, ˆ = δ or ) Ι () where Ι is the unit matri and the Kronecker smbol δ δ = if i= j and δ = 0 if i j. The answer is, H = H What is the Mises stress in terms of the deviatoric stresses? are its components, i.e., 3 = ˆ ˆ () noting that the summation convention applies to both indices. So the Mises stress does not depend upon the hdrostatic stress, onl upon the deviatoric stresses.
What is the phsical interpretation of the Mises ield criterion? An interpretation for isotropic media is that the Mises criterion is equivalent to a criterion for which ielding initiates when the elastic strain energ associated with distortion, i.e. with deviatoric strains, reaches a critical value. Specificall, the elastic distortion energ densit,ξˆ, is related to the Mises stress b ˆ + ν ξ =. 3E What are Mohr s Circles? Mohr s circles are a means of visualising the state of stress at a point. A vector field at a point can be visualised as an arrow. The Mohr s circles are the equivalent for the stress field. It is more complicated because stress is a second rank tensor rather than a vector. The Mohr s circles are drawn on a plot of shear stress () against direct stresses (). The centre of a Mohr s circle alwas lies on the -ais. How do ou draw the Mohr s circle for plane stress? Place the two principal stresses on the -ais and draw the circle through them, and with its centre at their mid-point. shear maimum shear τ A Y θ X direct stress B So what s it good for? The magnitudes of the two direct stresses and the shear stress in a coordinate sstem which is rotated b an angle θ from the principal aes are given b the points A and B. Note that the diameter AB is at an angle of θ to the -ais. The -coordinates of A and B are the two direct stresses, and, and the -coordinate of A is the shear stress. (Note that the -coordinate of B is an equal and opposite shear stress). From this it is readil seen that the stresses at an angle θ to the principal aes are, = ( + ) + ( ) cos θ = ( + ) ( ) cos θ (3) τ = = ( ) sin θ
These are consistent with the equations given above for a general rotation of coordinates from (, ) to (, ), but for the special case that the initial coordinate sstem is the principal sstem. Where is the point of maimum in-plane shear? Maimum shear must place A at the point with the largest -coordinate, i.e., at the top o of the circle. Hence, the maimum shear is obtained b rotating such that θ = 90, i.e., b 45 o from the principal aes. Moreover, the maimum shear is seen from the Mohr s circle to be just τ ma = ( ), as we have alread seen. The Mohr s circle shows that the direct stresses in this coordinate sstem are both equal to ( + ). What point on the Mohr s circle represents the state of stress at a point? It s a trick question. It s the whole Mohr s circle which represents the state of stress at a point. But then what do A and B represent? The diametrall opposite points A and B represent the components of the stress tensor with respect to a specific coordinate sstem, namel the coordinate sstem at an angle θ to the principal aes. The difference between the whole Mohr s circle and the points A,B is the same as the difference between the stress tensor and the stress matri. The stress matri gives the components of the tensor with respect to a given coordinate sstem. So, Mohr s circle stress tensor Points A,B stress matri
What are the Mohr s circles for a 3D state of stress? Now there are three principal stresses, so we can draw three different Mohr s circles according to which pair of principal stresses we choose to define the circle. What we get is, shear maimum shear possible impossible 3 impossible impossible direct stress possible Again this shows that the maimum shear is τ ( ) ma = 3. A rotation of the coordinate sstem in an one of the three principal planes moves the stress components around the relevant circle. But a general 3D rotation is not in one of the principal planes. For a general 3D rotation the stress components acting on an element of area are given b a point within the region between the inner and outer circles (marked possible ). The points within the inner circles, or outside the outer circle, cannot be reached, i.e., there is no orientation of an area element such that the area element is acted upon b such a combination of direct and shear stresses. What point on the 3D Mohr s circles represents the state of stress at a point? Not falling for it a second time, eh? You think that the whole of the three circles represents the state of stress at the point? Good tr, but no. It is the whole of the possible region between the inner and outer circles which represents the state of stress, i.e., the stress tensor. The correspondence in 3D is, The region between the inner & outer Mohr s circles stress tensor An points in this region the direct and shear stresses on an element of area
How do I find the point on the 3D Mohr s circles corresponding to a given orientation of the element of area? 3 Orientation of element of area (area normal) θ β These are the principal aes Objects versus Components Tensors, including vectors, are mathematical objects in their own right independent of the coordinate sstem used to describe them in terms of components. Object Components Vector Arrow ( v v vz ) Stress Tensor Mohr s Circles z z z z zz