CROSS-REFERENCE TO CO-PENDING PROVISIONAL APPLICATION  
       [0001]    This application claims priority from provisional application serial No. 60/229,416 entitled “Multiaxial Fatigue Crack Initiation in Rubber”, filed on Aug. 31, 2000, the entire contents of which are expressly incorporated by reference herein. 
     
    
     
       BACKGROUND OF THE INVENTION  
         [0002]    The present invention relates to the materials arts. It particularly relates to the analysis and estimation of fatigue life limited by crack formation and crack growth in materials undergoing applied stresses, especially rubber materials, and will be described with particular reference thereto. However, the invention will also find application in the analysis of other types of structural defects, and is furthermore applicable to materials other than rubbers that are undergoing mechanical stresses.  
           [0003]    Models for predicting fatigue life in rubber follow two basic approaches. One approach focuses on predicting crack initiation life, given the history of at-a-point quantities such as stress and strain. The other approach, based on ideas from fracture mechanics, focuses on predicting the propagation of a particular crack, given the energy release rate history of the crack.  
           [0004]    Several researchers have applied at-a-point quantities for life predictions in tires and other rubber parts. The quantities investigated have included maximum principal strain or stretch, maximum shear strain, octahedral shear strain, and total strain energy density. For incompressible materials, the total strain energy density is the same as the deviatoric strain energy density. These approaches generally assume that a unique relationship exists between the strain energy density and crack initiation life. While many in the rubber industry have used strain energy density as a predictive parameter for fatigue life, the range of validity of this approach under conditions typically experienced by parts in service has not been adequately investigated.  
           [0005]    Fatigue life analysis methods based on fracture mechanics typically presuppose the existence of an initial “test” crack and estimate its propagation under the strain history using iterative finite element analysis methods. Fatigue life is estimated by repeating the finite element analysis for a large plurality of test cracks with different sizes and orientations which are representative of the initial flaws believed to be present in the material. This approach is computationally expensive because each potential failure mode (i.e., test crack) requires its own finite element mesh and analysis. Furthermore, the crack propagation approach requires a priori knowledge of the initial location and state of the crack that causes the final failure. Often, this information is not available, and indeed is the very information the designer needs to predict.  
           [0006]    The present invention contemplates an improved fatigue life estimation method which overcomes the aforementioned limitations and others.  
         SUMMARY OF THE INVENTION  
         [0007]    According to one aspect of the invention, a method for estimating fatigue life for a material is disclosed. A multiaxial strain cycle is received that is described by a strain tensor that is a function of time. A hyperelastic constitutive model corresponding to the material is received. A fatigue crack growth curve is obtained. A cracking energy density is calculated based on the constitutive model and the multiaxial strain cycle. The cracking energy density is a function of material plane and indicates the portion of the total elastic strain energy density that is available to be released on a selected material plane. A cracking plane is determined based upon the cracking energy density. A fatigue life is estimated based on the cracking plane and the fatigue crack growth curve.  
           [0008]    According to another aspect of the invention, a method for identifying a cracking plane in an elastic material under the action of a tensile multiaxial strain history is disclosed. A cracking energy density W C  is calculated for a material plane. The cracking energy density is incrementally defined by, 
           
         dW 
         c 
         ={overscore (σ)}·d{overscore (ε)} 
       
           [0009]    with 
           {overscore (σ)}=σ {overscore (r)}={overscore (r)}   T   σ, d{overscore (ε)}=dε{overscore (r)}   
           [0010]    where dW c  is the incremental cracking energy density, σ is the stress tensor, ε is the strain tensor, and {overscore (r)} is a unit vector normal to the material plane. Calculating the cracking energy density W c  is repeated for a selected set of material planes. The cracking plane is identified based on the cracking energy density calculations.  
           [0011]    According to yet another aspect of the invention, a program storage medium is readable by a computer and embodying one or more instructions executable by the computer to perform a method for estimating a fatigue life in a material that undergoes a strain history. The method includes the steps of defining a plurality of spatial planes that collectively represent the material planes; calculating cracking energy densities corresponding to the plurality of spatial planes wherein the cracking energy density of a spatial plane indicates the energy available to propagate a crack in that spatial plane; and estimating the fatigue life based on the calculated cracking energy densities.  
           [0012]    Numerous advantages and benefits of the present invention will become apparent to those of ordinary skill in the art upon reading the following detailed description of the preferred embodiment. 
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0013]    The invention may take form in various components and arrangements of components, and in various steps and arrangements of steps. The drawings are only for the purpose of illustrating preferred embodiments and are not to be construed as limiting the invention.  
         [0014]    [0014]FIG. 1 schematically shows the quantities involved in calculation of the cracking energy density.  
         [0015]    [0015]FIG. 2 schematically shows a method for calculating fatigue life that suitably practices an embodiment of the invention.  
         [0016]    [0016]FIG. 3 schematically shows another method for calculating fatigue life that suitably practices another embodiment of the invention. 
     
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS  
       [0017]    With reference to FIG. 1, the cracking energy density (CED) is defined. The CED is the portion of the total elastic strain energy density (SED) that is available to be released on a given plane, e.g. through the nucleation or growth of a crack or other heterogeneous defect. The CED is defmed with respect to a spatial material plane  10 , which in FIG. 1 includes an exemplary crack  12  embedded in a material A. It will be appreciated by those skilled in the art that real materials (even those which appear macroscopically perfect) contain a certain volume density of heterogeneous structural defects, such as microscopic cracks, microscopic holes, foreign particles, or the like. It is also known that these structural defects typically have a certain size distribution that is characteristic of the material. In the art these intrinsic defects are often characterized by an initial flaw size that is considered an intrinsic property of the material. Thus, the exemplary crack  12  can represent a typical micro-crack of the virgin material having an initial flaw size characteristic of that material.  
         [0018]    The cracking energy density (CED) is defined in differential form as: 
           dW   c   ={overscore (σ)}·d{overscore (ε)}   (1) 
         [0019]    where: dW c  is the CED differential; {overscore (σ)} is the traction vector  14  onto the plane  10 , i.e. 
         {overscore (σ)}=σ {overscore (r)}={overscore (r)}   T σ  (2) 
         [0020]    wherein σ is the stress tensor and {overscore (r)} is the plane normal  16 ; and d{overscore (ε)} is the strain vector differential  18  associated with the plane  10 , i.e. 
         [0021]    ti  d{overscore (ε)}=dε{overscore (r)}   (3). 
         [0022]    where dε is the strain differential. By combining equations (1) through (3) and transforming into the principal coordinate system using a transformation κ, the CED differential or increment can be written as: 
           dW   c   ={overscore (r)}   T κ T   {overscore (σ)}d{overscore (ε)}κ{overscore (r)}   (4). 
         [0023]    In equation (4), the quantities σ and dε have been replaced by {overscore (σ)} and d{overscore (ε)} to indicate that these quantities are now written in terms of the principal coordinates.  
         [0024]    Both the SED and the CED are independent of strain path for elastic materials. Thus, the CED is found by integrating from the unstrained state to the strain state of interest ({overscore (ε)}) along any convenient path. The CED (W c ) can be written in integral form as:  
               W   c     =       r     -   T              κ   T          [       ∫   0     ɛ   _              σ   _                          ɛ   _           ]          κ          r   _     .               (   5   )                               
 
         [0025]    The expressions of equations (1) through (5) apply for both rotating and non-rotating tensile strain histories. In equation (5), the expression inside the square brackets contains terms describing the energy density contributed by each stress/strain increment pair in principal coordinates. The expression in brackets will be called the principal strain energy density matrix herein. The strain energy density (SED) is the trace of this matrix. The spatial plane comes into the CED in equation (5) through the plane normal vector {overscore (r)} in combination with the transformation κ outside the square brackets. Thus, it is seen that the CED calculation includes (1) calculating the strain energy density matrix which is independent of the choice of spatial plane, and (2) calculating the CED from the strain energy density matrix, wherein the CED can be viewed as the fraction of the total strain energy density that is available to be released on the plane corresponding to the plane normal {overscore (r)}.  
         [0026]    The general equations (1) through (5) which are generally applicable to any material undergoing a tensile strain history are applied in exemplary fashion to an isotropic, linear elastic material. For such a material, the principal stresses are related to the principal strains as:  
                 {           σ   1               σ   2               σ   3           }     =           2      G       1   -     2      v              [           (     1   -   v     )         v       v           v         (     1   -   v     )         v           v       v         (     1   -   v     )           ]            {           ɛ   1               ɛ   2               ɛ   3           }         ,           (   6   )                               
 
         [0027]    and the principal strain energy density matrix is written as:  
                 ∫   0     ɛ   _              σ   _                          ɛ   _           =                        2      G       1   -     2      v                      [               (     1   -   v     )          ɛ   1   2       +     v                   ɛ   2          ɛ   1       +     v                   ɛ   3          ɛ   1             0       0           0             (     1   -   v     )          ɛ   2   2       +     v                   ɛ   3          ɛ   2       +     v                   ɛ   1          ɛ   2             0           0       0             (     1   -   v     )          ɛ   3   2       +     v                   ɛ   1          ɛ   3       +     v                   ɛ   2          ɛ   3               ]     .                 (   7   )                               
 
         [0028]    It is emphasized that the equations (1) through (5) are general in nature, and equations (6) and (7) give a particular embodiment thereof.  
         [0029]    With reference to FIG. 2, a method  30  for estimating the fatigue life of a material undergoing a particular tensile strain history  32  is described. The method  30  is not restricted with respect to the material type, but is limited to non-rotating strain histories.  
         [0030]    In addition to the strain history  32 , the method  30  receives as inputs a hyperelastic stress-strain relationship  34 , a fatigue crack growth curve  36 , and values for the initial crack size and the critical crack size  38 . The critical crack size is the size at which the flaw or crack becomes sufficiently large to terminate the useful life of the product comprising the material. The crack growth curve  36  is optionally represented by a complex curve that is numerically implemented using piece-wise integration. However, it is common practice in the art to assume a power-law behavior of the fatigue crack growth curve  36  over the life of the crack or flaw from initial size to critical size. Under this assumption, it is often found that the fatigue crack life is independent of the critical crack size.  
         [0031]    The method  30  uses an iterative approach to identify the material plane on which the crack causing the failure will occur. For a non-rotating strain history, this plane is the plane for which the CED is greatest. Thus, a discrete number of material planes to be analyzed are defined in a step  44 . These material planes comprise a discrete search space that is searched using an iterative loop  46 .  
         [0032]    After selecting a material plane in a step  48 , the CED is calculated for the selected plane in a step  50 . The calculating step  50  evaluates equation (5) for the selected plane, using the strain history  32  and the hyperelastic stress-strain relationship  34  as inputs. It will be appreciated that the method  30  does not involve finite element analyses of the type used in prior art methods of iteratively searching for the cracking plane.  
         [0033]    After looping  46  through the material planes, the cracking plane is identified in a step  54  by selecting that plane having the largest CED. This selection is valid for non-rotating strain histories  32 .  
         [0034]    With the cracking plane identified in the step  54 , the time required for a crack to grow from its initial size to the critical size can be calculated by any convenient means. Advantageously, this calculation takes advantage of the CED calculated for the cracking plane in the step  50 . In the method  30 , a strain energy release rate is calculated in a step  56  from the CED, and the time interval for crack growth from initial to critical sizes is calculated in a step  58 . This interval corresponds to the fatigue life  60 .  
         [0035]    Fatigue crack growth in many materials, including rubber, is driven by the strain energy release rate, even under complex strain histories. The strain energy release rate cannot, in general, be computed from the total strain energy density. The strain energy release rate varies proportionally, for small cracks, with the CED and the crack size. The cracking energy density represents the portion of the total strain energy density that is available to be released on a given material plane. For a non-rotating strain history, crack initiation occurs on the plane that maximizes the cracking energy density, so that the cracking plane can be identified simply by analyzing the CED, e.g. as in step  54 .  
         [0036]    With reference to FIG. 3, a method  70  is disclosed which is applicable in the general case including rotating strain histories. The method  70  again takes as inputs a multi-axial strain cycle or strain history  72  (which in the method  70  can include rotating strain histories), a material constitutive model or hyperelastic stress-strain relationship  74 , a fatigue crack growth curve  76 , and values for the initial crack size and the critical crack size  78 . Spatial planes representative of the material planes are defmed in a step  84 , and a loop  86  processes each material plane in turn. A material plane is selected in a step  88 , and a CED is calculated for the plane using the strain history  72  and the hyperelastic stress-strain law  74  as inputs.  
         [0037]    Because the correspondence between the plane having the maximum CED and the cracking plane is not strictly valid for general (e.g., rotating) strain histories, the method  70  computes the strain energy release rate  92  and the time interval for the crack growth to reach critical size  94  for every material plane, i.e. the computations  92 ,  94  are included within the loop  86 , and the entire loop  86  is repeated for all the material planes in a step  96 . With the actual time interval for crack growth to the critical size calculated for each material plane in the loop  86 , the fatigue life is obtained in a step  100  by selecting the minimum time interval for a crack to grow to the critical size.  
         [0038]    Although the method  70  is more computationally costly versus the method  30 , it is still much faster than the prior art methods which repeated finite element analyses for each of the material planes.  
         [0039]    The invention has been described with reference to the preferred embodiments. Obviously, modifications and alterations will occur to others upon reading and understanding the preceding detailed description. It is intended that the invention be construed as including all such modifications and alterations insofar as they come within the scope of the appended claims or the equivalents thereof.
A multiaxial strain cycle ( 32, 72) is received that is described by a strain tensor that is a function of time. A hyperelastic constitutive model ( 34, 74) corresponding to the material is received. A fatigue crack growth curve ( 36, 76) is obtained. A cracking energy density is calculated ( 50, 90) based on the constitutive model ( 34, 74) and the multiaxial strain cycle ( 32, 72). The cracking energy density is a function of material plane ( 44, 84) and indicates the portion of the total elastic strain energy density that is available to be released on a selected material plane ( 48, 88). A cracking plane is determined ( 54, 98) based upon the cracking energy density. A fatigue life is estimated ( 60, 100) based on the cracking plane and the fatigue crack growth curve ( 36, 76).